What do Catholics believe about Yom kippur and judgment?

Christ's New Law has abrogated the Old Law's ceremonial precepts (if Yom Kippur is even a ceremonial precept of the Old Law or an invention of Talmudic Judaism, which is a different religion than Judaism before the destruction of the Temple in 70 A.D.). St. Thomas Aquinas Summa Theologica I-II q. 103 a. 3 ("Whether the ceremonies of the Old Law ceased at the coming of Christ?") c.:

All the ceremonial precepts of the Old Law were ordained to the worship of God as stated above (Question [101], Articles [1],2). Now external worship should be in proportion to the internal worship, which consists in faith, hope and charity. Consequently exterior worship had to be subject to variations according to the variations in the internal worship, in which a threefold state may be distinguished. One state was in respect of faith and hope, both in heavenly goods, and in the means of obtaining them—in both of these considered as things to come. Such was the state of faith and hope in the Old Law. Another state of interior worship is that in which we have faith and hope in heavenly goods as things to come; but in the means of obtaining heavenly goods, as in things present or past. Such is the state of the New Law. The third state is that in which both are possessed as present; wherein nothing is believed in as lacking, nothing hoped for as being yet to come. Such is the state of the Blessed.

In this state of the Blessed, then, nothing in regard to worship of God will be figurative; there will be naught but "thanksgiving and voice of praise" (Is. 51:3). Hence it is written concerning the city of the Blessed (Apoc. 21:22): "I saw no temple therein: for the Lord God Almighty is the temple thereof, and the Lamb." Proportionately, therefore, the ceremonies of the first-mentioned state which foreshadowed the second and third states, had need to cease at the advent of the second state; and other ceremonies had to be introduced which would be in keeping with the state of divine worship for that particular time, wherein heavenly goods are a thing of the future, but the Divine favors whereby we obtain the heavenly boons are a thing of the present.

Since the Old Law has been abrogated by the New, it follows that to the extent that Yom Kippur was prescribed by the Old Law, there is no requirement either to celebrate it or to hold any particular belief concerning it; whereas if it was not prescribed by the Old Law, the same is true a fortiori.

It is not the practice of the Catholic Church to have doctrines or precepts about particular Jewish Holy Days or celebrations. If the Church did, it would be easily found in the Catholic Encyclopedia. Here is what the Catholic Encyclopedia actually has to say about Yom Kippur.

The rites to be observed on the Day of Atonement are fully set forth in the sixteenth chapter of Leviticus (cf. Exodus 30:10; Leviticus 23:27-31, 25:9; Numbers 29:7-11). It was a most solemn fast, on which no food could be taken throughout the whole the day, and servile works were forbidden. It was kept on nineteenth day of Tischri, which falls in September/October. The sacrifices included a calf, a ram, and seven lambs (Numbers 29:8-11). But the distinctive ceremony of the day was the offering of the two goats.

"He (Aaron) shall make the two buck-goats to stand before Lord, in the door of the tabernacle of the testimony: and casting lots upon them both, one to be offered to the Lord and the other to be the emissary-goat: That whose lot fell to be offered to the Lord, he shall offer for sin: But that whose lot was to be the emissary goat he shall present alive before the Lord, that he may pour out prayers upon him, and let him go into the wilderness . . . After he hath cleansed the sanctuary, and the tabernacle, and the altar, let him offer the living goat: And putting both hands upon his head, let him confess all the iniquities of the children of Israel, and all their offences and sins, and praying that they may light on his head, he shall turn him out by a man ready for it, into the desert. And when the goat hath carried all their iniquities into an uninhabited land, and shall be let go into the desert, Aaron shall return into the tabernacle of the testimony." (Leviticus 16:7-10, 20-23) - Catholic Encyclopedia.

The Church does however uses Melchisedech, Yom Kippur, Manna, and the Paschal Lamb to represent the Eucharist in one fashion or another.

The Church recognizes many and various realities from the Old Testament as figures for Christ’s gift of himself in the Most Holy Eucharist. On the Solemnity of Corpus Christi in the Ordinary Form, the Church read from the book of Exodus – how the people were cleansed and the covenant ratified through animal sacrifice at the foot of Mount Sinai.

There are so many images and figures for the Eucharist in the Old Testament – the Manna, the bread and wine offered by Melchisedech, the sacrifice of the Day of Atonement (Yom Kippur), the Paschal Lamb, etc.!

Yom Kippur and the sacrifices of the Old Law

"But Christ, being come an high priest of the good things to come, by a greater and more perfect tabernacle not made with hand, that is, not of this creation: Neither by the blood of goats, or of calves, but by his own blood, entered once into the holies, having obtained eternal redemption." (Hebrews 9:11-12)

While all the sacrifices of the Old Law point to the perfect sacrifice which Christ would offer in his own blood upon the Cross, that expiatory sacrifice of Yom Kippur (the Day of Atonement) is the chief figure of the Crucifixion of our Savior.

On this day, the only time in the year, the high priest would enter into the Holy of Holies and, taking the blood of sacrifice, would sprinkle this blood in the direction of the Ark of the Covenant. Further, the blood would be sprinkled in several other places in the Temple. The priest, on this single day, invoked the Most Holy Name of the Lord – seeking mercy for himself and for the people.

This was the only day of the year in which anyone entered the Holy of Holies, which was the most sacred part of the ancient Temple. This Holy of Holies was a sign and figure for the eternal sanctuary of heaven, which Christ opened through the shedding of his own Most Precious Blood. - Melchisedech, Yom Kippur, Manna, and the Paschal Lamb: Figures of the Eucharist

The Association of Hebrew Catholics has this to say about celebrating Jewish feasts and holidays:

As Hebrew Catholics, we can recognize this same teaching technique in the parables of Jesus and the Sacraments which He initiated. Bread and wine, water and oil, incense and candles, prayers and songs are all rich and integral parts of Catholic liturgy and tradition.

By celebrating the Hebrew feasts in light of Catholic truth, we can pass on to our children the great heritage of Faith, the wonder of God’s actions in the history of His people, and the enduring promise of salvation through Yeshua haMashiach.

Here is what they say about Yom Kippur:

Yom Kippur, the Day of Atonement, is the day when God’s judgment is believed to be sealed for the coming year. The ten days between Rosh Hashonah and Yom Kippur are called the Days of Penitence. It is a time to repent of and to correct those things we have done wrong, especially to our neighbor, and to ask God’s forgiveness.

For everyone thirteen years and older, Yom Kippur is a day of fasting. Fasting encourages humility and repentance, disciplines the body and clarifies the mind as we examine our conduct and thoughts.

In the synagogue, the ancient Kol Nidre (Absolution of All Vows) is chanted and confession of sins as commanded in Leviticus 26:40 is made. At home, memorial candles are lit for members of the family who have died. The Mourner’s Kaddish and final blowing of the Shofar conclude the day.

I have continued to keep Yom Kippur as a day of fasting and repentance, not for myself but for the Jewish people, especially those who have turned away from God. I also light candles and have Masses offered for departed relatives. It is the only holiday that makes itself known by the absence of food and festivities. We offer our prayers as we conclude the fast at the evening meal. - Yom Kippur

The Association of Hebrew Catholics has the ecclesiastical support of Archbishop Raymond L. Burke.