Wonju Airport is physically in Hoengseong-gun. Korea Airports Corporation lists the address as 38 Hoengseong-ro, Hoengseong-eup, Hoengseong-gun, and the airport's own homepage now labels the airport 'Hoengseong/Wonju.' That detail matters when you plan a hotel, a train connection or another regional bus.
The official directions page currently lists local buses 1 and 13, plus bus 2 between Hoengseong Intercity Bus Terminal, Wonju Airport, Wonju Station and Wonju city, and bus 2-1 between Hoengseong Intercity Bus Terminal, the airport and Wonju's intercity/express bus terminal. Do not save those numbers without rechecking the travel day: local schedules can change, and the correct route depends on which Wonju hub you actually need.
For a taxi fallback, the stand is outside Arrival Gate 2. The current airport estimate is about 5 minutes and KRW 5,000 to Hoengseong Intercity Bus Terminal, about 30 minutes and roughly KRW 19,000-20,000 to Wonju's express/intercity terminal, and about 35 minutes and KRW 32,000 to Wonju City Hall. The airport notes that actual taxi fares vary with traffic.
Before flying, identify the exact station, terminal or accommodation in Korean and plan the full route from Hoengseong. If you are connecting onward to another Gangwon destination, compare whether the Hoengseong side or a Wonju transport hub gives the simpler connection rather than automatically going into central Wonju.