I would just reïterate that opening hours will be perturbed that day, and information you can find on websites are not necessarily to be relied upon, unless New Year’s Eve is specifically addressed. For instance, the Vatican Museum is normally open on the last Sunday of the month, but it will be closed on the day of your visit, which is in fact a last Sunday. It may therefore not be easy to discover ahead of time what will be open. Moreover, you don’t necessarily want to spend your time standing in a queue waiting to get into St. Peter’s Basilica or the Pantheon. With that I mind, I might recommend a wholly outdoors walking tour of the historic centre, beginning at the Coliseum and ending in Trastevere.

You might, for instance, get the Leonardo Express to Termini and then the Metro to Colosseo, and then follow the route shown in the maps below to St. Peter’s Square and then on to Trastevere. It might take some 3 or 4 hours to cover, depending on your pace, hitting many of the major sights/sites of Rome. You might even pop into some of the churches along the route, e.g., Dodici Apostoli, Santa Maria del Popolo, San Luigi dei Francesi, Santa Maria sopra Minerva, Santa Maria in Trastevere, if mass is not in progress and there is no queue to get in, perhaps even the Pantheon. In Trastevere, you should be able to find a restaurant open for an early lunch, after which you would take Tram 8 to the Trastevere railway station, and then the local train back to the airport.