There are no direct busses from Kuching International Airport to the town center. However, it will take only a 15-20 minute walk to reach the nearest public bus station. We picked up this map from the tourist information counter at the airport:
All busses traveling in the indicated direction will stop in Kuching center.
If you want to go back from Kuching center to the airport, you don’t have to take the walk. Upon request, the bus will take a detour and drop you off directly at the airport for a small extra fee.
A timetable for the public busses can be picked up from the Visitor’s Information Centre at the main bazar (Kuching waterfront). But be aware that usually the busses go 10 to 30 minutes before the scheduled time! (E.g., the park officer at Kubah National Park advised us: „The 3:30 bus goes at 3 o’clock, so be at the bus stop 10 minutes to 3“)