Should be right by the climbing shops (which as of last year were along the Chonggye stream, which made me think they were/are going to be priced out soon). . . some are in the interior, so follow the narrow aisles. I bought a down bag in Korea about 8 years ago and it's pretty good. Obviously, if you're buying here, your best deals will be local brands-- not uber cheap (like Chinese rip-offs) but decent quality. For foreign brands, order online and have it shipped to the military.
Sleeping bags are usually rated by degrees, rather than seasons, like tents. They are not so versatile, as in you need a summer and winter bag (I'm up to 2 each). Down is used for winter, as it is very warm and packs down well, but it's not easy to wash and you shouldn't get it wet. Synthetic winter bags are also available, but these tend to get clumpy with use, and don't pack down as well. You'll have to think how you'll use it, and really you only need down if you go winter camping! Then it's great. I have a 0 degree down and it's more than I've ever needed, inluding several days on Rainier, winter camping in Korea, etc. Still, for that down you have time. . . don't think you'll need it anymore in Korea for the season!
"If you can't do something well, you might as well learn to enjoy doing it poorly."
-- from a de-motivational poster, but I find it oddly liberating