Looking for Asian Markets in Austin
MT Market is very large and has a tremendous variety (mostly Chinese). For Korean try Han Yang Market or Dong Yang Market (New Oriental Market), both on Lamar near Airport Blvd; my wife who is Korean prefers Han Yang for most food items except kimchi, and Dong Yang for kimchi and the eat-in cafe.
MT Market is the biggest by far (bigger than many of our regular grocery stores) and should have just about anything you'd need if you're cooking Chinese, Vietnamese, or Korean. It's located in the Chinatown Center at North Lamar and Kramer.
They devote a decent amount of shelf space to Japanese goods as well, but it's in their version of an International section. if they don't have it, Asahi Imports on Burnet Rd might. They aren't quite as cheap though.