With my poor Korean skills I found the following information, I am not very sure so any better answer is welcome.
- Go to http://www.airport.co.kr
- Click 전국공항 네트워크 (national airports), or the hamburger menu if on a small screen
- Click 무안국제공항 (Muan International Airport)
- In the 교통/주차 menu, click 대중교통 (public transport)
- Scroll to the bottom of the page, there is a table of bus lines.
- Find the row where the second column reads 무안국제공항 → 목포버스터미널 (Muan International Airport → Mokpo Bus Terminal) and click the arrow at the right
- All scheduled times appear, find the last one. Currently it is:
11회(심야) : 무안국제공항(22:30) ☞ 무안버스터미널(22:45) ☞ 청계버스터미널(22:55) ☞ 목포버스터미널(23:10)
11th bus (at night): Muan International Airport (22:30) ☞ Muan Bus Terminal (22:45) ☞ Cheonggye Bus Terminal (22:55) ☞ Mokpo Bus Terminal (23:10)
Direct link: http://www.airport.co.kr/muan/extra/bus/busList/layOut.do?langType=1&menuId=988
The price seems to be 5200 won.
The page also links to the official web page of the destination bus station: https://www.usquare.co.kr/kor/terminal/mokpo.do
Update: I actually took that last bus
My plane was late, but I was among the first passengers to exit the plane. I got out of immigration at 22:29 (foreigners queue on the right, takes a long time for each passenger), rushed through customs, ran to the exit (good thing I had no checked-in luggage), turned right at the exit, they were two buses waiting:
- One for Gwangju (광주시) on the right
- One for Mokpo (목포시) on the left
The bus driver understood "Mokpo", I jumped in.
The bus leaved at 22:32, as 90% of the passengers were still lining up at immigration.