Alice, I'm not sure if you've cruised out of other Florida ports, but I think you'll find Tampa's cruise port a much more tourist-friendly place from which to begin your voyage. You definitely want to arrive a day ahead to allow for train delays, but showing up "too early" is never a problem as there are multiple entertainment, shopping, and dining options available in and near the port. The Florida Aquarium, SS American Victory Museum Ship, and Channelside complex are literally steps away from your ship.
Tampa Union Station is a great point of entry for cruisegoers. Arriving only blocks away from the port, your transfers both before and after the cruise will be a breeze compared to those of your shipmates arriving to Tampa by air!
Union Station lies between the port and the Ybor City Historic District, which has some very convenient hotel options. Ybor's flavor and atmosphere is reminiscent of New Orleans (less cajun... more cuban) and there are numerous dining and nightlife venues withing walking distance of Ybor hotels. Unless you're bringing a large volume of luggage, the TECO Line Streetcar is a cheap, convenient, and fun way to get from Ybor's hotels right to the port for embarkation.
Hotels near Channelside and the convention center also offer great convenience to the port area.
Good luck in planning your trip. Going from "rail" to "sail" should be a great vacation. Bon Voyage!