Paul's Epistles are always written to churches who have already heard the gospel and the story of Jesus. They are written to encourage and correct those congregations. There is no need for him to repeat every detail of his initial Gospel teaching in every epistle. That would be redundant.

As far as the claim that Paul knew no details of Christ's life:

23 For I have received of the Lord that which also I delivered unto you, that the Lord Jesus the same night in which he was betrayed took bread:

24 And when he had given thanks, he brake it, and said, Take, eat: this is my body, which is broken for you: this do in remembrance of me.

25 After the same manner also he took the cup, when he had supped, saying, this cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me. 1 Corinthians 11

This is a specific and direct counter-example to the premise. Paul here refers to the what he and the Corinthians already knew. This reinforces the fact that Paul and the churches he planted were familiar with details from Christ's earthly ministry.

Paul clearly knew about the life of Jesus, as he affirms this in his own words in many places:

1. Paul knew the words spoken by our Lord Jesus

   Acts 20:35: I have shewed you all things, how that so labouring ye ought to support the weak, and to remember the words of Jesus, how He said, It is more blessed to give than to receive

2. Paul knew about the Communion service and what had happened that night

   2 Corinthians 11:23-25 : I have received of the Lord that which also I delievered unto you, That the Lord Jesus the same night in which He was betrayed took bread: and when He had given thanks, He brake it, and said, Take eat: this is my body, which is broken for you: this do in remembrance of me. After the same manner also He took the cup when He had supped saying, This cup is the new testament in my blood: this do ye, as oft as ye drink it, in remembrance of me

3. Paul knew about John the Baptist's mission - this clearly shows that he knew the current events that had occured in his time.

   Acts 13:24: When John had first preached before his coming the baptism of repentance to all people of Israel. And as John fulfilled his course, he said, Whom think ye that I am? I am not he. But, behold there cometh one after me, whose shoes of his feet I am not worthy to loose.

4. Paul knew about the death of Jesus, what caused it, how He was executed and His resurrection

   Acts 13:27-31 For they that dwell at Jerusalem, and their rulers, because they knew him not, nor yet the voices of the prophets which are read every sabbath day, they have fulfilled them in condemning him. And though they found no cause of death in him, yet desired they Pilate that he should be slain. And when they had fulfilled all that was written of him, they took him down from the tree, and laid him in a sepulchre. But God raised him from the dead: And he was seen many days of them which came up with him from Galilee to Jerusalem, who are his witnesses unto the people.

5. Paul knew what the prophets, and Moses had written about Christ. We can see from the Gospels multiple instances where it is quoted "as written by prophet". So Paul clearly knew about all such events

   Acts 28:23 And when they had appointed him a day, there came many to him into his lodging; to whom he expounded and testified the kingdom of God, persuading them concerning Jesus, both out of the law of Moses, and out of the prophets, from morning till evening.

All emphasis in italics and bold have been added and are not present in the Bible

It's important to note that the video in question is a presentation of the Christ myth theory, which is a fringe academic theory. Very few scholars give it any credence; in fact, New Testament scholar Larry Hurtado places Christ myth theorists in the same category as flat earthers and moon landing deniers. The theory's flawed methodology leads the narrator in the video to make the ludicrous claim, contrary to clear evidence, that Paul "never quotes anything that Jesus is supposed to have said."

But even among less extreme skeptics of Christianity, there is often a claim that, because Jesus' biography was not written down quickly and was embellished in the meantime, we know very little of the truth. Since Paul's letters are dated significantly earlier than the four canonical gospels and don't give much of a biography of Jesus, such skeptics frequently try to enlist his letters to demonstrate that even the earliest Christians knew almost nothing of the life of Christ.

In the following, I attempt to give a tour of Paul's letters to demonstrate his familiarity with the fleshly, earthly Jesus in a (hopefully) skeptic-friendly way; scholars debate whether Paul wrote all the letters attributed to him and many scholars don't believe that Acts quotes Paul accurately, so I've italicized references to such passages where Paul's authorship is disputed, and bracketed claims where there is no citation from the undisputed writings.

Paul's biography of Jesus, reconstructed

Jesus is both divine (Rom 1:4; Rom 9:5; 1 Cor 8:6, 10:4; Phil 2:6; Col 1:15-20; 1 Tim 3:16; Tit 2:13) and a Jewish (Rom 9:5; Gal 3:16, 4:4) descendant of David (Rom 1:3, 15:12; 2 Tim 2:8; Acts 13:22-23). [His ministry was foreshadowed by that of John the Baptist. (Acts 13:24-25)] He had many apostles, who were miracle-workers (2 Cor 12:12), including his brother James, John, Peter, and the twelve (1 Cor 9:5, 15:5-8; Gal 1:17-19).

He lived sinlessly (2 Cor 5:21), and then after his last meal (1 Cor 11:23-26) certain Jews caused his death (1 Thess 2:14-15; Acts 13:27). Following a Roman trial (1 Tim 6:13; Acts 13:27-28), he died by crucifixion (1 Cor 1:23, 2:2; 2 Cor 13:4; Gal 3:1; Phil 2:8; Col 2:14; Eph 2:16), in accordance with God's will (Rom 5:6-8; Gal 1:4) and with prophecy (1 Cor 15:3; Acts 13:29), as a sacrifice (Rom 3:25, 8:3; 1 Cor 5:7).

After his death, he was buried and then raised (Phil 3:10) by God (Rom 4:24, 10:9; 2 Cor 4:14, 13:4; Eph 1:20; 2 Tim 2:8) on the third day (1 Cor 15:4,20), and appeared to the twelve and to hundreds more (1 Cor 15:5-8; Acts 13:30-31). He is now at the right hand of God (Rom 8:34; Phil 2:9; Col 3:1; Eph 1:20; 1 Tim 3:16), as the Messiah and Lord (too many verses to count).

Further comments

While I was preparing the reconstruction above, I consulted the work others who have undertaken similar projects, including Bart Ehrman (from his book Did Jesus Exist?), Mark Goodacre (Did Paul Think That Jesus Was the Pre-existent Son of God? and What Did Paul Know About Jesus?), Craig Evans (Jesus Tradition in Paul's Letters), F.F. Bruce (Paul and the Historical Jesus), and Bob Seidensticker (What Did Paul Know About Jesus? Not Much). Seidensticker is a blogger and a Christ myth theorist, Bruce and Evans are evangelical scholars, and Ehrman and Goodacre are New Testament scholars in the academic mainstream. You'll notice, if you follow the links, that my preceding summary is substantially similar to each of theirs. Unsurprisingly, Seidensticker is the most significant exception; his conclusion that "Paul doesn't even place Jesus within history" can be easily rejected, and his conclusion that "the Jesus of Paul isn't the Jesus of the gospels" shouldn't be accepted without a lot more scrutiny. Instead, Bruce's conclusion does a better job of cohering with the evidence:

What Paul has to say of the life and teaching of the historical Jesus agrees, so far as it goes, with the outline preserved elsewhere in the New Testament and particularly in the four Gospels.

Nevertheless, I must also agree with Ehrman (see more here):

Imagine what we wouldn't know about Jesus if these letters were our only sources of information. We hear nothing here of the details of Jesus' birth or parents or early life, nothing of his baptism or temptation in the wilderness, nothing of his teaching about the coming Kingdom of God; we have no indication that he ever told a parable, that he ever healed anyone, cast out a demon, or raised the dead; we learn nothing of his transfiguration or triumphal entry, nothing of his cleansing of the Temple, nothing of his interrogation by the Sanhedrin or trial before Pilate, nothing of his being rejected in favor of Barabbas, of his being mocked, of his being flogged, etc. etc. etc. The historian who wants to know about the traditions concerning [the life of] Jesus - or indeed, about the historical Jesus himself - will not be much helped by the surviving letters of Paul.

However, it seems reasonable to imagine that if Paul thought his letters would be the final word on the life of Jesus, he would have taken greater care to describe it in more detail. And for billions of Christians, his writings represent only about a third of the corpus of the New Testament, while the Gospels represent about half. So Christians have never relied exclusively on Paul for such details.

Faulty Premise: Paul knew nothing of Jesus

Paul summarizes his own life story in Galatians 1:

11 I want you to know, brothers and sisters, that the gospel I preached is not of human origin. 12 I did not receive it from any man, nor was I taught it; rather, I received it by revelation from Jesus Christ.

13 For you have heard of my previous way of life in Judaism, how intensely I persecuted the church of God and tried to destroy it. 14 I was advancing in Judaism beyond many of my own age among my people and was extremely zealous for the traditions of my fathers. 15 But when God, who set me apart from my mother’s womb and called me by his grace, was pleased 16 to reveal his Son in me so that I might preach him among the Gentiles, my immediate response was not to consult any human being. 17 I did not go up to Jerusalem to see those who were apostles before I was, but I went into Arabia. Later I returned to Damascus.

18 Then after three years, I went up to Jerusalem to get acquainted with Cephas and stayed with him fifteen days. 19 I saw none of the other apostles—only James, the Lord’s brother. 20 I assure you before God that what I am writing you is no lie.

By his own testimony, Paul had at least two weeks' direct contact with Peter (aka Cephas) and James, two obvious eyewitnesses of Christ. Even if you discount the personal encounter with Jesus in Acts 9, Paul clearly had access to those who did know Christ.

Indeed, in his time, Paul was being accused not of not knowing who Jesus was, but rather of trying to claim the other disciples' experiences of his own. Hence Paul going out of his way in vs. 19 to distance himself from the apostles...

When he says earlier that the Gospel he preaches is not of human origin, he is referring to his direct encounter with Christ, through whom we surmise he learned of Christ. Later, Peter, who clearly knew Christ, attests that the encounter was genuine and that this is the same Gospel.

To say Paul knew nothing of Christ simply does not square with the primary sources.