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Thank you all for taking time to answer my question in great detail, especially Mr Aldrich. I will look into those shuttle websites as to me it's the safest option.
The cheapest option is to rent a car one-way from the airport, drive to the port, drop luggage/family off at port, drop rental car off and take rental car shuttle back to the port. Repeat in reverse for your post-cruise trip back to the airport. Budget, Thrifty, Avis and then Hertz all have drop off locations and free round trip shuttles very close to the port (companies are listed in order of closeness to the port). These four places are open on Sundays and their port shuttles start early in the morning until the afternoon. Cost approx $45.
After car rental, the next recommended alternative to a taxi is a shuttle service.
Shuttle companies mentioned on Cruise Critic:
Cocoa beach Shuttle 321-631-4144 or 888-784-4144; www.cbshuttle.com
Art’s Shuttle 321-783-2112 or 800-567-5099; www.artsshuttle.com (Mention Cruise Critic for 10% discount)
Note: Most of the popular shuttle companies servicing the port are not private shuttles (they will wait for all reservations to board before leaving). If you need a private shuttle request it specifically!
Just getting a cab in the street will cost $120.
Ask royal caribean if they have a ships bus for transfer.
Try CityOneExpress, $22 per person one way, I think it's cheaper if you book a return journey.
Further to Mr Moss's suggestion RCL will only offer a transfer if you fly into Orlando on the day the cruise departs - and I understand from Mr Pearsons original question is that they will be there already.
Disney run shuttle buses to Port Canaveral, as their ships leave fron there too. Sue.
You should use Cityone xpress Transportation because they provide very professional and affordable service. The wed site is "www.cityonexpress.com".