That is a tourist agency run route. So first you must reserve by phone call. That bus is running every weekend. They will run bus in week, when passenger is 15 people.
To depart is
05:05 am at south gate of DSME
05:07 am in front of Ocean plaza
05:09 am in front of Okpo Kookmin bank.
And it will arrive in Muju resort at 08:00 am.
To depart in Muju resort is at 16:20 pm.
I'm not a partaker of tourist agency. I saw that web page and I'm telling you about that now.
Perhaps, if you wonder more, you write a comment here.