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I'm on mobile and don't have full access to the Miami-Dade Transit site, but I seem to remember that there is a bus line that serves the port and takes you over the bridge to the main part of downtown Miami, where you can pick up the bus to the airport (from memory Route 7 and/or J might be the route that goes all the way from downtown Miami to the airport.
Edit to update:
Looks like it is Route 243 that serves the cruise port area - however, this is a weekday-only route, only between 06:00 to 09:00 and then again from 15:00 to 19:00. Apparently there is no service on weekends:
http://www.miamidade.gov/transit/alignment.asp#243