I've used Shuttle Service to Port Canaveral Cruise Lines: portcanaveralshuttleservice.com ~ Rates
before. Pickup from airport was very fast and friendly. Benefit of this one is the more people going the less you pay per person as many charge a flat fee.
If you want peace of mind most if not all of the lines now offer this service to and from the airport although they tend to charge a bit more and it is a flat fee per person. For 1-2 that's cheap, if you have 8 people you'd be better off with your own van in an outside company. Also they will hold the ship if your running late or fly you to the next port in the off chance you don't make it on time.
Airport is quite far from the port so taxi is not likely going to be your best option unless you arrive late and need to get to the port fast.