We did exactly what you are talking about and it worked out great, with the exception of renting from Budget after the cruise.
We flew into Orlando and rented from
Avis. Great experience, upgraded 1 car class and on our way within a short time. It took maybe an hour from the airport to get to the Radisson. Very straight shot, nothing fancy. Checked into the Radison - highly recommend it. Great pool, food within walking and driving distance, decent room, etc. Turned in the car the next morning and took the shuttle over to the port. We didn't even inquire into the hotel shuttle but from talking to other people, it booked really fast. The Avis shuttle was first come, first serve starting around 10 a.m., but we didn't have to wait.
Once we returned there were Avis and Radisson vans waiting for the ship when we got to the area they were staging transportation at. In fact, an Avis van took us and another couple to Budget b/c the Budget van was taking so long to get back. I highly recommend Avis due to the fact that they have at least 2 vans along with the Radisson vans going back and forth. They all work together so it doesn't matter which van you end up on. If we do it again we will definitely rent from Avis in both directions.
One of the highlights for us was meeting other people on our cruise at the Radisson the night before. We chatted while our kids swam. Then hung out some while on the cruise and at
castaway cay.
If you have any other questions.... just ask!
-joy