We arrived off Rhapsody of the Seas 2 years ago at around 4 p.m.. We ended up walking out to the "southernmost point in the U.S.," hoping to find some museums (the lighthouse, too) to visit along the way, but most were closing or closed. We hired 2 bicycle "cabs" to take us out to Ft. Zachary Taylor State Park--not a great beach, but the kids had fun, and we stayed to see the sunset from there, and it was gorgeous--and not crowded at all, compared to the crowds in the square in town. We had met a park ranger and he told us some places to eat and get some key lime pie--we took a taxi from the park to the restaurant, had a wonderful meal (I think we were the only cruise ship passengers there), then took our 2 teens back to the ship. My husband and I then went in search of the elusive key lime pie, walked around town, then back to the ship.
Unfortunately, arriving into Key West so late in the day didn't leave much time to see things--not too many ship's excursions, either (the trolley?) The main places still open were souvenir shops, restaurants and bars. The ship didn't depart until 1 a.m. or so, so there was plenty of time for those things if you're interested. Hopefully you're on a ship that gets in early in the day---heard the trolley tour and the Mel Fisher museum were good.