Most car rental companies will have you drop the car off at their Miami airport location. The only exception to this that I can think of is Hertz, which has a downtown location with free shuttle to the port. Alamo and Thrifty also have free shuttles, but these are from the Miami airport.
If you are going to do the rental car option, use one of these 3 companies to save the $25 cab fare from MIA to the port. I have also heard stories that representatives from Budget and National will advise over the phone that they have a shuttle to the port, but they do not. You safest bet is to stick with Hertz (to downtown), or Thrifty or Alamo (to MIA).
And to answer another question...yes you can drop luggage off at the ship, then return your car.
Fascination 9/2004
Triumph 11/2005