Like JohninDC said the hotels should be opening up soon. Check every day or twice a day. The Hilton portfolio of Hotels usually show an outrageous price the first few days after opening. Then they have gone down to a more reasonable price. Nov or so the busy season starts so the hotels and prices fill up/rise pretty fast.
We usually rent from National at 608 N Nebraska Ave. This is a cobranded location with Alamo and Enterprise I believe. No shuttle to/from port that I know of. We usually stay at The Hampton Inn/ Home2Suites at 1155 E Kennedy. It is less than a 10 min walk to the National location I mentioned. They have a free cruise shuttle that will drop you off and pick up. You just have to call them for pick up from the ship. I can give you the number if needed. We have walked to terminal 6 and 3, and used shuttle from 2. There is also a Publix grocery,CVS drugstore, restaurants and free Teco Trolley close by.
The other closest are Avis at Embassy Suites on S Florida Ave, Hertz at Marriott on S Florida also. Enterprise also has a location at 1001 W Kennedy. Dont know about any shuttles with these. Just watch opening and closing times esp(Sat/Sun) with these locations. They arent 24hr like some airport locations.
Like JohninDC also said the Tampa Airport 20 min away from port is also a possibility.
Dave