There will be restaurants that are open, especially in the city center, so you can have dinner out (most restaurants open at 7 or 7;30pm for dinner). You can also have lunch when you arrive, but many restaurants close for lunch after 2 or 2:30 pm to then reopen later for dinner. Coffee bars should be open at any time. Lot's of snack bars/pizza snacks options in the city center as well.
Grocery stores will be largely closed. There might be some open in the historical center. Usually stores of the Carrefour and Conad chains may have some open locations with limited hours. But I'm sure after a long flight you don't want to cook. Just go and eat out at a restaurant.
Museums should be open, churches should be open, walking outside will be allowed. Just be aware that the walk to San Miniato is uphill (but downhill coming back to the center).