We sailed out of Miami a couple of years ago and stayed the night before we set sail at the Holiday-Inn Port of Miami. It was a decent hotel ... no complaints from us. It is right across the street from the Bayside Market so we spent the evening walking around there.
As for rental car agencies we rented a car from National on our return to Miami. I was told that they picked up at the Port. I checked before our cruise and was told that National picked up at the Port. When we got off the ship the RCCL staff told me that National picked up at the port. After waiting for 30 minutes and not seeing any National buses I finally called them only to be told that no they do not pick up at the port.... we just took a cab. I do recall several other car rental buses passing us including Avis and Hertz. That was a couple of years ago so I would check with them again.