Fredericton Transit only operates within city limits. The Lincoln bus turns around at the last intersection in the city, and that's still a good 5-6 km away. Never heard anyone even suggest extending service to the airport.
Saint John does have an airport route, though it only runs 12 hours a day, 5 days a week:
http://www.saintjohntransit.com/cont...ps/Route32.pdf