We will have two full days in Rome, December 31 and January 1. We will need to get the touristy stuff done on December 31. What is open (or never closed) that we can see on January 1?
Pretty much everything is closed so you might want to just walk around and enjoy the open-air sights such as Piazza Navona with its fountains, Jewish Ghetto, Circus Maximus, etc. People jump off a bridge into the Tiber River on New Year's Day, I think around 11, so a walk along the river upstream from Tiber Island could be entertaining. In the northern part of the city, the Borghese Park will be open and many Roman families head there. There is a little lake with rowboats to rent (also entertaining even if you don't row, as in my experience Italians are really bad at rowing).
Ostia Antica seems to be open on December 31.
I can't find a current version of this list from the Rome Office of Tourism, so the info is for the 2022-2023 holiday season, but it's actually quite encouraging:
You can check the websites of the places that were open on New Year's Day 2023 to see if the same policy will hold in 2024.
We walked around the city on Jan.1 looking for a place to buy lunch. The only place we found open was a small Middle Eastern place. We sat outside in the sun eating their fabulous pizza.
Hotel restaurants should be open to take care of their guests so check them out.