At Cheongju International Airport, choose your onward mode before leaving the terminal. Korea Airports Corporation currently says regional/intercity buses board from stops in front of the passenger terminal. The taxi stand is across the road from Gate 2. Cheongju Airport railway station is not inside the terminal: the airport’s own guidance places it about 600 meters from the intercity-bus terminal area.
That distinction matters with heavy luggage, rain, heat or a tight connection. If the rail timetable suits your destination, the station can be useful, but do not treat it like an airport subway station with a platform under the terminal. For destinations in Cheongju or elsewhere in Chungcheong, compare the current bus timetable with the rail option before committing to the walk.
For a late arrival, check the final bus and train before landing. If neither timing works, use the official taxi rank rather than assuming a rail departure will still be available. Korea Airports Corporation lists customer service at 1661-2626 and publishes current airport transport information.