pick your poison, if you've got a harness you may want to get on the ropes. Seoul's geography is really interesting, with small mountains throughout the city, with some pretty reasonable approaches to mostly granite slab crags. Plenty of artificial outdoor walls as well, if your climbing partner has a rope and draws you can hit up pretty much any of these walls for free.
Most of the gyms in Seoul are bouldering basements, with I think three main exceptions:
K2 near Kongkuk university station: I had a mixed experience at this gym, and can't say it was really worth the price of admission, though this was about 9 months ago. I didn't touch their vertical wall, though.
The North Face Climbing Center at Suyu station. I am a member here, and I wrote the gym entry. Love this place, and they'll lend you a rope.
and
O2 World (Kolon Sports?): I haven't been to this place, and i don't know how accurate the gym info still is, but I hear it's big and nice.