"Does the New Testament predict Jesus' return on Rosh Hashanah?"
Yes it does, but not in the way many Christians expect.
The "coming" referred to in the New Testament was fulfilled in the events just preceding the Jewish war, and which then allowed the Temple to be destroyed.
Jesus' ministry brought about the end of the Old Covenant. We see him fulfill the annual feasts not only in his ministry, death, resurrection and ascension, but in a greater level which includes the "firstfruits" church as his body.
His death fulfilled and transformed Passover into Christian communion. His resurrection and ascension fulfilled Firstfruits, after which he sent the promised Spirit at Pentecost. (Notice that 3000 were saved, rather than the 3000 slain at the first Pentecost).
What many Christians don't get is that the Feast of Trumpets was fulfilled in the legal Covenant witness of the apostolic church, gathering a new Israel as an army of holy martyrs. (Revelation itself refers to the "ten days" of testing in 2:10, five and ten being military numbers in the Bible).
This witness of "Trumpets" led to the fulfillment of the Day of Atonement (Day of "Coverings"), during which Judah herself was finally divided in two: those who received Christ's High Priesthood and those who rejected him. Matthew 23-25 predict this time. The sins of those who believed were covered (as the first goat) and the sins of those who did not, in fact, the blood of all the prophets beginning with Abel fell upon that generation.
All of the Old Testament saints ascended into heaven. And this is the event predicted by Paul in Thessalonians: the first resurrection. Abraham and all the saints received their heavenly country. Josephus records "chariots in the clouds" circling the city.
Once "Jerusalem below" was destroyed, and circumcision finished (at least in God's economy), the Feast of Tabernacles was fulfilled in the completion of a priesthood of all nations (referred to in Zechariah 14:6).
This festal process is also the deep structure of the Revelation. The seven trumpets are the apostolic witness. Jerusalem is referred to as Egypt, Sodom and Babylon. As Egypt, the slaughter of the innocents would be avenged. As Sodom, once the saints had been removed from the city, she would be burned with fire and left sterile, childless, forever. As Babylon, her "sun, moon and stars" would fall, to be replaced by the saints, resurrected and shining (Daniel 12).
The Revelation also uses symbols from Jericho, where priestly trumpets brought down her walls. Israel was "re-circumcised" outside the walls preceding this liturgical warfare, then the priests led the "circumcision of the city" for seven days. Likewise, the apostolic church was circumcised "without hands" as the true Jews (Jews indeed, not just in word, fulfilling the "stoicheia" of Judaism forever). The Romans dug a trench around the city, then built a second wall, crucifying up to 500 escapees from the siege per day. So "the circumcision" whose false doctrine the apostles fought against got what they desired. The Romans had moved in when leading Jews from all over the empire, the hard-liners, were in the city. Rejecting Christ as their Passover, there was "no more sacrifice for sins," and "their table became a snare." Trapped in the city, it seems that a contingent of Jews were converted, possibly remembering the words of Jesus, and were slain. This is what is spoken of in Revelation 14, a "firstfruits" offering ascending to God as the first goat of atonement, leaving the city without any righteous, fully exposed to the wrath of God, with no covering, but "calling the rocks and hills" to cover them.
This means that since AD70, there is no longer in God's eyes either Jew or Gentile, only believers and unbelievers in Christ. The Old Covenant was finished forever. Being a Jew was not only racial, but also an office before God. That office was decommissioned and replaced in the first century. When the Jewish rulers rejected the Spirit of God after Pentecost, Judaism was filled with even worse demons, as Jesus predicted. Which is why the Revelation speaks of the judgment of the harlot for her (Jewish) sorceries. Sorcery in the Bible is always an attempt to gain the blessings of obedience to the ethics of God's Covenant without the actual obedience. Hence Paul speaks of "obedience to the gospel."
So the last trumpet was the final warning to the Jews, most likely the witness of the book of Hebrews and the testimony of the last apostle, whose "seven thunders" brought down the seven bowls, the curses of Moses, upon Israel according to the flesh, who, unlike Jesus, died "in the wilderness" because she accepted Satan's offer of kingdom before God's time.
"Truly, I say to you, there are some standing here who will not taste death until they see the Son of Man coming in his kingdom.” (Matthew 16:28)