Here's some places you might start; I translate as "wall" but it may be a gym. . .
Hyundai Art Hall Artificial Climbing Wall
Hanbaek Alpine Club Climbing Wall
in Ok-dong, by Gunbu University
I think there's another place, too. I am pretty sure one or more advertises in magazine Climber
. . . if you still can't find something when I'm back in Korea I can do a further search or you (could have someone) contact the Ulsan Alpine Federation
e-mail : firstname.lastname@example.org
/ phone. 052-296-8848 / fax. 052-296-8516