There are many options for transportation and shuttles between the Tampa airport and the cruise terminal. You can catch a shuttle, hop in a taxi, rent a limousine, or rent a car. It should take about 25 minutes to go the 5 miles from the airport to the port.
Shuttles and taxicabs are available at the Tampa cruise terminal and at the airport. A list of transportation companies that serve the port can be found on this Tampa ground transportation page. Contact numbers are provided for taxis, shuttles, limousines, rental cars, and public transportation. It also has the minimum flat rate taxi fares.
There are so many options for Tampa airport to cruise port transportation that you can't go wrong. Just pick a one and go.