Disney has not yet sailed out of Port Everglades in Fort Lauderdale, this will be a new cruise terminal once operations begin this fall.
That being said...
I flew into Fort Lauderdale for a Miami cruise (the night before!) and we stayed at Hyatt House right there at the airport. Well appointed, affordable, and free shuttle from airport to hotel. We then used
Jiffy Jeff Transportation to get us from the hotel to the Port of Miami. They do also service Port Everglades, and I enjoyed our driver and their service enough that I would use them again.
Having said that, I would personally also look into booking transfers directly with Disney Cruise Line if they are offering them. You might need to get the complementary shuttle form the hotel back to the airport to board DCL buses, but doesn't sound too bad either. Price comparison would factor into my choice here