There is a surprisingly large number of bars in Muju - unfortunately, none of them serve any imported liquors or beers whatsoever. Basically, the menu is limited to one brand of soju, and three brands of Korean beer.
3040.
For a nice, quiet atmosphere, try 3040. The owner here speaks some English, and one bottle of beer will cost you 2500 won.
Tudari.
If interested in a decent selection of Korean food to go with your drinks, try Tudari. Another chain establishment, it's slightly more expensive than some of the local options, but has an excellent selection of food. 4000-10,000 won food, 3000 won beer.