Yes, many museums and churches are closed -- but if the day is nice, you can most certainly take advantage and enjoy the day outdoors. You can walk around the center and enjoy the architecture and views of the buildings, you can cross Ponte Vecchio and walk up to Piazzale Michelangelo for the panoramic view of the entire city, you can take the bus up to Fiesole and see the view from up there.
Many restaurants and other food places will be open, so don't worry about finding a place to eat at ;-)
Worry more about for the night before to book a restaurant ahead, since on the night of 31st many restaurants decide to do a fixed menu and limited seating and close the doors to have everyone celebrate together -- so it will be a tad more difficult to find a place to eat dinner on the 31st.