Opening this week at Cinecube is the 2005 film The King.
Starting Tuesday is the Taiwnese Film Festival at the Seoul Cinematheque, featuring films by Hou Hsaio-hsien, Edward Yang, Ang Lee, Tasi Ming-Liang and others. The schedule is here. Nothing on the site yet about English subtitles.
There is the Seoul International Film Festival from June 5-11. The schedule and description of the films is here.
And the Seoul LGBT festival, whose website is here. Unfortunately, the English section of the site does not seem to have any information.