Look at the WikiVoyage page for Cairns under "Get In", it mentions that there's a nearly-hourly airport shuttle running into the city centre. From other comments, I gather that the shuttle company mentioned in the wiki voyage article isn't the only one running.
However, the shuttle bus website linked from WikiVoyage is a rather broken site. There's no timetables there, no information, and the online reservation form doesn't work either... That's also apparently not the only one.
That leads me to wonder - is it possible to just arrive at the airport, find the bus stop(s), pay and get on the next bus?
Or do I need to pick one company, phone them up and try to book by phone onto one specific bus?