There are some bars in Korea that are considered "Western bars." These bars are where there is a slight chance you actually might run into another Westerner. They do often times play more Western music and they oftentimes have English menus.
There are a few of these bars on what we call "the bar street" (the street probably has a name).
Here are some photographs and a videos of a few of the Western bars last December. In particular The 501 Club, R & B and the Mini Bar.