On this
link you can see the list of 'N' night buses to the airport, which run's roughly after midnight to about 5am.
Most of the regular CityFlyer airport buses start up again right after 5am. Depending where you're coming from, I think you can still catch the first CityFlyer service in the morning and get to the airport at or around 6-615am.
If you take the Airport Express, it starts at 6am so you'll get to the airport at 6:24am.