With my poor Korean skills I found the following information, I am not very sure so any better answer is welcome.

1. Go to http://www.airport.co.kr
2. Click ???? ???? (national airports), or the hamburger menu if on a small screen
3. Click ?????? (Muan International Airport)
4. In the ??/?? menu, click ???? (public transport)
5. Scroll to the bottom of the page, there is a table of bus lines.
6. Find the row where the second column reads ?????? ? ??????? (Muan International Airport ? Mokpo Bus Terminal) and click the arrow at the right
7. All scheduled times appear, find the last one. Currently it is:

11?(??) : ??????(22:30) ? ???????(22:45) ? ???????(22:55) ? ???????(23:10)

Translation:

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 (???) on the right
• One for Mokpo (???) 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.