Yes, many Christians have reported that Jesus Christ has appeared to them. It is not always entirely clear whether the appearance is to be understood as a physical appearance or a vision.

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Reputable

The matter of a "reputable" individual presents a challenge. The skeptically-minded can simply answer the OP with a resounding "no", by questioning the reputation of anyone who is named. This is the oldest trick in the book. Every prophet & apostle in the Bible about whom we have any decent set of biographical data was questioned, challenged, and considered by many to be disreputable.

In my previous response I shared two of the appearances of Christ to Joseph Smith, which are widely known, regularly taught, and considered to be of immense theological significance to members of The Church of Jesus Christ of Latter-day Saints. In this response I will catalogue a more lengthy list of experiences that meet the following 3 criteria:

• I am reasonably familiar with the people/story (there may be countless stories of which I am not aware)
• I consider the people to be of good character & sound mind
• Their story has been shared publicly. Let us remember that a face to face experience with the Savior would be one of the most sacred experiences of a person's life, and is not something to be treated lightly. Many do not share such experiences at all, let alone publish them on the internet.

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The question of Apostolic Commission

Although an apostle has the responsibility to be a witness of the resurrection of Christ, it is apparently not the case that all individuals (even just in the NT) who beheld the risen Christ were commissioned as apostles.

For example, there is no evidence that Joanna (Luke 24:10) was ever an apostle. Paul reports that Jesus appeared to more than 500 people at once (1 Cor. 15:6), yet Paul's authoritative use of the title "apostle" strongly suggests there were not 500+ apostles on the earth at that time (note that I am using "apostle" in the Acts 1:25-26 sense of the word).

Ergo, that one sees the risen Christ does not mean one is necessarily commissioned as an apostle.

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Individuals

1. Joseph Smith reported that He was called by God to restore the fulness of the Gospel of Jesus Christ to the earth. As part of his prophetic instruction & revelatory responsibilities, he reported that the Savior appeared to him on several occasions. The 3 best known are:

• Spring 1820 ("The First Vision"), wherein the Father & the Son appeared to him in answer to prayer regarding which church to join (see verses 15-20 here)
• February 1832, Joseph Smith & Sidney Rigdon beheld the kingdoms of glory, saw Jesus on the right hand of the Father, and heard the witness that Jesus is the Only Begotten of the Father (see verses 23-24 here)
• April 1836 at the Kirtland Temple - the Lord appeared to Joseph Smith & Oliver Cowdery, accepted the temple, and sent messengers to restore a variety of priesthood keys to the earth (see verse 2-8 here)

2. Sidney Rigdon - 1832 - see section on Joseph Smith

3. Oliver Cowdery - 1836 - see section on Joseph Smith

4. Zebedee Coltrin (and numerous others)(Coltrin was a friend & student of Joseph Smith) - 1833:

At one of these meetings after the organization of the school, (the school being organized on the 23rd of January, 1833), when we were all together, Joseph having given instructions, and while engaged in silent prayer, kneeling, with our hands uplifted each one praying in silence, no one whispered above his breath, a personage walked through the room from east to west, and Joseph asked if we saw him. I saw him and suppose the others did and Joseph answered that is Jesus, the Son of God (see here)

*editorial note--I have been in the room where this occurred

5. John Murdock (a friend & student of Joseph Smith) - 1833, also present in the meeting described by Zebedee Coltrin:

the eyes of my understanding were enlightened, and I saw the form of a man, most lovely, the visage of his face was sound and fair as the sun. His hair a bright silver grey, curled in most majestic form, His eyes a keen penetrating blue, and the skin of his neck a most beautiful white and he was covered from the neck to the feet with a loose garment, pure white, whiter than any garment I have ever before seen. His countenance was most penetrating, and yet most lovely. (see here)

6. Lorenzo Snow (5th President of The Church of Jesus Christ of Latter-day Saints) - 1898 - as reported by his granddaughter Allie Young, to whom he related the account:

This special evening Allie was with her grandfather in his office. The doorkeepers had gone home and the night watchmen had not yet come in, so they were alone. When Allie was ready to leave, President Snow went to a dresser and took a large bunch of keys from the drawer so that he could let her out the main entrance. Together they walked down a large corridor near the celestial room.

President Snow suddenly stopped and said, “Wait a moment, Allie. I want to tell you something.” Allie listened intently as her grandfather told her of an unforgettable experience he had once had at that place in the temple: “It was right here that the Lord Jesus Christ appeared to me at the time of the death of President Woodruff. He instructed me to go right ahead and reorganize the First Presidency of the Church at once and not wait as had been done after the death of the previous presidents, and that I was to succeed President Woodruff [as President of the Church].”

President Snow held out his left hand and said, “He stood right here, about three feet above the floor. It looked as though he stood on a plate of solid gold.”

Still speaking in hushed, reverent tones, President Snow told Allie that the Savior’s appearance was so glorious and bright that he could hardly look at Him.

President Snow put his right hand on Allie’s head and said, “Now granddaughter, I want you to remember that this is the testimony of your grandfather, that he told you with his own lips that he actually saw the Savior, here in the temple, and talked with him face to face.” (source)

7. Orson F. Whitney (an apostle of The Church of Jesus Christ of Latter-day Saints) - prior to 1926:

I shall never forget the kind and gentle manner in which He stooped and raised me up and embraced me. It was so vivid, so real that I felt the very warmth of His bosom against which I rested. Then He said: ‘No, my son; these have finished their work, and they may go with me; but you must stay and finish yours.’ Still I clung to Him. Gazing up into His face—for He was taller than I—I besought Him most earnestly: ‘Well, promise me that I will come to You at the last.’ He smiled sweetly and tenderly and replied: ‘That will depend entirely upon yourself. (source)

8. Rebecca Bean (missionary for The Church of Jesus Christ of Latter-day Saints) - prior to 1933

Out of respect for Sister Bean's momentous experience, related at length, I will simply offer a link to her story, and share one statement the Savior made to Sister Bean, in commendation of the service she was providing to the church's missionaries:

I promise you if you will go about your work as you have done it this day you will be equal to it. Oh, remember these missionaries represent me on this earth, and all that you do unto them you do unto me.

Her story as a missionary, and that of her husband Willard Bean, are presented in the based-on-a-true-story film The Fighting Preacher.(highly recommended, and it includes a relation of the experience discussed above)

9. Mark E. Petersen (an apostle of The Church of Jesus Christ of Latter-day Saints) - before 1973:

Paul was one of the great witnesses of Christ anciently. We have many witnesses for Christ today. I am one of them, and I humbly stand before you as one of his witnesses (source)

10. Hugh B. Brown (an apostle of The Church of Jesus Christ of Latter-day Saints) - 1970s - two accounts, related here, including the statement (recorded by his nephew):

He said it was not a vision, but the Lord appeared to him

See also accounts in I Know He Lives: How 13 Special Witnesses Came to Know Jesus Christ

Bonus Material

The OP originally referenced the appearance of Christ to Paul on the road to Damascus. The New Testament records multiple appearances of the Savior after this time:

11. To Paul in Acts 23:11

12. To John on Patmos (Revelation 1:13-18). There's no scriptural or historical argument that 1 Cor. 15 describes the last ever appearance of the Savior. Both #11 & #12 (and indeed #s 1-10) occurred after 1 Corinthians was written.

Saint Peter.

Off of the top of my head, aside from Paul and the author of the Book of Revelation, Saint Peter, the first Pope, also had an encounter with Jesus following the latter's ascension into Heaven.

The way the story goes, Peter fled the city of Rome because the Romans had a warrant out for his arrest and execution, but on the road out of Rome, he encountered Jesus, who was walking into Rome. He asked Jesus where he was going, and Jesus responded "To Rome, to be crucified again." Peter then proceeded to turn around and return to Rome, where he was executed by crucifixion.

Joseph Smith claimed that the Lord Jesus Christ appeared to him on more than one occasion. His statements regarding two of those experiences are provided below.

Spring 1820 near Palmyra, New York, God the Father and His Son Jesus Christ appeared to Joseph Smith:

I saw a pillar of light exactly over my head, above the brightness of the sun, which descended gradually until it fell upon me. It no sooner appeared than I found myself delivered from the enemy which held me bound. When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other—This is My Beloved Son. Hear Him! (Joseph Smith History 1:16b-17)

April 3, 1836 in Kirtland, Ohio, the Savior appeared to Joseph Smith & Oliver Cowdery together:

2 We saw the Lord standing upon the breastwork of the pulpit, before us; and under his feet was a paved work of pure gold, in color like amber.

3 His eyes were as a flame of fire; the hair of his head was white like the pure snow; his countenance shone above the brightness of the sun; and his voice was as the sound of the rushing of great waters, even the voice of Jehovah, saying:

4 I am the first and the last; I am he who liveth, I am he who was slain; I am your advocate with the Father. (Doctrine & Covenants 110:2-4)

That Joseph Smith would be considered disreputable by some was predicted exceptionally well--from Joseph's Smith's personal history:

God had a work for me to do; and that my name should be had for good and evil among all nations, kindreds, and tongues, or that it should be both good and evil spoken of among all people. (Joseph Smith History 1:33b)

How true that statement is.

Has any reputable Christian claimed to have been physically visited by Jesus Christ Himself after Paul's conversion?

Ecclesiastical history has many cases of such claims. The real problem here is how to distinguish the seemingly real ones from those that are held to be true according to legend or tradition.

The Catholic Church admits to Jesus visiting people on many occasions, usually in the lives of saints. Even here some are well documented, while other visitations are now reduced to legend and/or tradition.

St. Stephen, the Protomartyr of Christians, like St. Paul had a vision similar to that of St. Paul. However, I am not sure this type of heavenly vision of Jesus is something you would be looking for, so I will simply mention it in passing. Besides,this happened prior to St. Paul's conversion

55 But he, being full of the Holy Ghost, looked up stedfastly into heaven, and saw the glory of God, and Jesus standing on the right hand of God,

56 And said, Behold, I see the heavens opened, and the Son of man standing on the right hand of God. Acts 7:55-56

St. Peter

Still within Apostolic Times, the Catholic Church traditionally that Jesus visited St. Peter while he was fleeing Rome, in order to avoid being martyred. There is a small church on the outskirts of Rome dedicated to this event.

Quō vādis? is a Latin phrase meaning "Where are you marching?". It is also commonly translated as "Where are you going?" or, poetically, "Whither goest thou?".

The phrase originates from the Christian tradition regarding Saint Peter's first words to the risen Christ during their encounter along the Appian Way. According to the apocryphal Acts of Peter (Vercelli Acts XXXV; late 2nd century AD), as Peter flees from crucifixion in Rome at the hands of the government, and along the road outside the city, he meets the risen Jesus. In the Latin translation, Peter asks Jesus, "Quō vādis?" He replies, "Rōmam eō iterum crucifīgī" ("I am going to Rome to be crucified again"). Peter then gains the courage to continue his ministry and returns to the city, where he is martyred by being crucified upside-down. The Church of Domine Quo Vadis in Rome is built where the meeting between Peter and Jesus allegedly took place. The words "quo vadis" as a question also occur at least seven times in the Latin Vulgate.

First documentary evidence for the church comes from the monastery of Sant'Alessio all'Aventino in the 9th century, and describes the church as ubi Dominus apparuit ("where the Lord appeared"). The circumstances of its actual foundation are unknown, but it is reasonable to surmise that it was in response to the legend in the Acts of Peter just quoted. - Domine Quo Vadis

As for other visitations of Jesus to people, according to Catholicism, it seems to be in the hundreds. What this comes down to, is where someone is willing to accept the information at face value or to reject it.

For example, let us take the visions of St. Margaret Mary Alacoque.

St. Margaret Mary Alacoque

In this monastery Alacoque received several private revelations of the Sacred Heart, the first on 27 December 1673 and the final one 18 months later. The visions revealed to her the form of the devotion, the chief features being reception of Holy Communion on the first Friday of each month, Eucharistic adoration during a "Holy hour" on Thursdays, and the celebration of the Feast of the Sacred Heart. She stated that in her vision she was instructed to spend an hour every Thursday night to meditate on Jesus' Agony in the Garden of Gethsemane. The Holy Hour practice later became widespread among Catholics.

On 27 December 1673, the feast of St. John, Margaret Mary said that Jesus had permitted her to rest her head upon his heart, and then disclosed to her the wonders of his love, telling her that he desired to make them known to all mankind and to diffuse the treasures of his goodness, and that he had chosen her for this work.

Initially discouraged in her efforts to follow the instruction she had received in her visions, Alacoque was eventually able to convince her superior, Mother de Saumaise, of the authenticity of her visions. She was unable, however, to convince a group of theologians of the validity of her apparitions, nor was she any more successful with many of the members of her own community, and suffered greatly at their hands. She eventually received the support of Claude de la Colombière, the community's confessor for a time, who declared that the visions were genuine. In 1683, opposition in the community ended when Mother Melin was elected Superior and named Margaret Mary her assistant. She later became Novice Mistress, and saw the monastery observe the Feast of the Sacred Heart privately, beginning in 1686. Two years later, a chapel was built at Paray-le-Monial to honor the Sacred Heart.

In 1689, Alacoque received a private request from Jesus to urge the King of France, Louis XIV, to consecrate the nation to the Sacred Heart, so that he may be "triumphant over all the enemies of Holy Church." Louis XIV, along with his successors, Louis XV & Louis XVI failed to consecrate the nation, and 100 years after Alcoque's vision, Louis XVI was stripped of his power in 1789, during the events of the French Revolution.

Alacoque died on 17 October 1690.

Then there is the vision of Jesus to St. Faustina Kowalska, the Apostle of Divine Mercy.

St. Faustina Kowalska

On February 22, 1931, Our Lord and Savior Jesus Christ appeared to this simple nun, bringing with Him a wonderful message of Mercy for all mankind. Sister Faustina tells us in her diary under this date:

“In the evening, when I was in my cell, I became aware of the Lord Jesus clothed in a white garment. One hand was raised in blessing, the other was touching the garment at the breast. From the opening of the garment at the breast there came forth two large rays, one red and the other pale. In silence I gazed intently at the Lord; my soul was overwhelmed with fear, but also with great joy. After a while Jesus said to me, 'paint an image according to the pattern you see, with the inscription: Jesus, I trust in You.'"

Some time later, Our Lord again spoke to her:

"The pale ray stands for the Water which makes souls righteous; the red ray stands for the Blood which is the life of souls. These two rays issued forth from the depths of My most tender Mercy at that time when My agonizing Heart was opened by a lance on the Cross....Fortunate is the one who will dwell in their shelter, for the just hand of God shall not lay hold of him." - Catholic Apparitions of Jesus to Blessed Faustina, Crakow, Poland: Devotion to the Divine Mercy

The original Image of the Divine Mercy, painted under the guidance of Saint Faustina.

One of the other nuns, Sister Christine, later stated that rays of light from the window were visible that night and attracted the attention of people standing on the other side of the street, implying that it was a "physical" appearance, rather than an interior vision.

I could go on and on giving example from the lives of saints, like of St. Thomas Aquinas himself to St. Anthony of Padua. It should also be be known that there are souls at every epoch in time that Our Lord is visiting somewhere in the world. For Catholics, this is somewhat normal.

Just as a side note, I would like to add the following observation as I find it somewhat interesting.

Myrna Nazzour

Some have claimed that Myrna Nazzour, has from December 1982 had apparitions of Our Lady and to a very limited way from Our Lord in the form of allocations. What is interesting here is that she is from Soufanieh, a suburb of Damascus in Syria. This is the very region of St. Paul’s vision of Jesus on his voyage to Damascus.

Myrna Nazzour

Ordinary people

I know of many situations of ordinary people seeing Our Lord as if in a vision, here and there. One in particular case was in my parish with a nine year old boy dying of cancer, some years ago. The day he died, he said that Jesus was standing at the end of the bed and asked him to join him in heaven. He then asked his mom for permission to die. She took him in her arms and she said yes. He immediately died.

The most clear example I'm aware of is the Hispanic preacher Yiye Ávila:

José Joaquín Ávila Portalatín (September 11, 1925 – June 28, 2013), better known as Yiye Ávila, was a Puerto Rican Pentecostal Evangelist and writer of Sephardic origins. His preaching and messages were characterized in proclaiming that the coming of Christ is imminent. He is considered one of the most influential Protestant preachers of the Spanish language. He was also the second runner-up for Mr. North America in 1953.

(source)

Yiye Ávila had a very powerful conversion testimony, as well as many powerful subsequent experiences throughout all his 50+ years of ministry.

I understand spoken Spanish pretty well, so I took the liberty to write down and translate (to the best of my ability) the transcript of the first couple minutes of this video. Judging by the setting and the quality, it looks like the video was recorded when Yiye was preaching in a jail a few decades ago.

Transcript (in Spanish):

No pasaron ni dos semanas, cuando una noche de madrugada yo fui despertado, y cuando desperté oí pasos de alguien que venía por el pasillo de mi casa caminando. Yo me quede atónito. ¿Qué es esto? Son como las 4 de la mañana y hay una persona caminando por el pasillo. Mi esposa estaba durmiendo al lado mío. Y de pronto la puerta de la habitación se abrió. Y cuando se abrió la puerta, entró aquel hombre, vestido de blanco, el pelo largo sobre los hombros, me miraba fijamente. Yo me quedé tenso. No podía mover un músculo. Se me pegó bien pegadito a mí, como para que entendiera que era real. Me metió las manos por debajo de mi cuerpo y me levantó de la cama como a un niño recién nacido. Me aguantó en sus brazos 20 - 30 segundos, y me bajó, me puso en la cama, sacó los brazos, se me quedó mirando y salió, y por el pasillo oía los pasos [makes sounds of footsteps], hasta que salió de la casa ... Por eso estoy aquí, en esta tarde. Para predicarles un Cristo real! No un Cristo que cuelga en una pared y se carga al hombro, de madera, que coge polilla. Un Cristo vivo, real! Que te ama, que te quiere salvar! Que quiere que tú seas de él, y darte una experiencia a ti personal con él, para que tu vida sea transformada! Ahí yo ya no titubeé más nada, se acabaron los titubeos. Se me acabaron los temores. Señor, lo que tú quieras! A los pocos días estaba en una iglesia predicando [...]

English translation (my best attempt):

Not even two weeks passed, when one night near dawn I was woken up, and when I woke up I heard the footsteps of someone who was walking down the hallway of my house. I was in shock. What is this? It's about 4 am and there is someone walking down the hallway. My wife was sleeping next to me. And suddenly the bedroom's door opened. And when the door opened, that man came in, dressed in white, long hair over his shoulders, staring at me. I got tense. I couldn't move a muscle. He got very close to me, so as to make me understand that He was real. He put his hands under my body and lifted me up out of bed like a newborn baby. He held me in his arms for 20 - 30 seconds, and then he put me back down in the bed, removed his arms, stared at me a bit longer and left, and I heard the footsteps down the hallway [makes sounds of footsteps], until he left the house ... That's why I'm here this afternoon. To preach a real Christ! Not a Christ that hangs on a wall and is carried on shoulders, of wood, that gets moth-eaten. A Christ that is alive, real! Who loves you, who wants to save you! Who wants you to be His, and give you a personal experience with Him, so that your life may be transformed! After that experience I doubted no more, doubts were over. Fears were over. Lord, I'll do whatever you want! Within a few days I was already in a church preaching [...]

If anyone is interested in a longer version of his testimony (1h 13m), here is another link.