I think in practice you would be wanting a dedicated service from the airport to the CBD, which kind of already exists in the form of the existing transfer operators and cabs. It isn't an enormous impost in terms of time or cost the way it is at SEQ airports. Diverting services from the northern suburbs via the airport is a bit much in my view.
With some creative rostering you might be able to create a simple connection from the CBD corridor (Cairns Central and the city bus station) up Lake St and straight into the airport which doesn't chew up too many resources.