I would prefer to get near the port after landing than deal with it the next morning. You can sleep a bit later, take your time getting to the ship, etc.

Look into a one way car rental too if you feel like driving, for us it was cheaper to pick up a Hertz at the Lauderdale airport, drop it off at First street or where ever Hertz is down there (don't remember the exact address) the next morning and use the Hertz shuttle to the cruise. It was considerably less expensive than the two cabs would have been.