Is the Post Office open on New Year's Day?

No.New Year's Day is a Federal holiday in the US.

In the United States the Postal Service does not operate on New Year's Day as it is a declared federal holiday. Most shipping services (UPS, FedEx, DHL) are also closed on New Year's Day. However, many retail services (malls, shops, etc.) are open on New Year's Day as many other businesses are closed to observe the holiday.

New Year's Day in the UK is a Bank Holiday and the Post Office is closed. If New Year's Day falls on a Saturday or Sunday then the Monday is taken as the Bank Holiday.