Muan International Airport sits in Muan-gun, not in central Gwangju or Mokpo. The current Korea Airports Corporation directions page identifies direct Kumho Buslines corridors between Gwangju U-Square and the airport, and between Mokpo Bus Terminal, Muan Bus Terminal and the airport.
The airport's public-transport page currently displays a Gwangju/Mokpo timetable labeled effective from December 1, 2024. Because that effective-date label is older than the present travel date, treat the displayed times as a current official reference page, not as proof that your exact bus will operate unchanged on the day you fly. Recheck the airport notice, bus operator or terminal before building a tight connection around it.
For Gwangju, the displayed route includes U-Square, Kimdaejung Convention Center and Gwangju Songjeong Station before the airport. For Mokpo, it includes Mokpo Terminal, Namak, Muan Terminal and the airport. The airport currently places the bus stop about 20 meters to the right after exiting first-floor Gate 1; the taxi stand is also outside Gate 1.
If your flight lands near the final bus, have a taxi fallback and know which city you are actually heading to. A traveler connecting to KTX at Gwangju Songjeong should plan differently from someone staying in Mokpo or transferring through Muan Terminal. Save the exact Korean hub name, recheck the final departure on the day, and leave extra buffer for flight delays.