Seoul is one of the best night-life cities in the entire world.

My suggestion is to get a hotel within a 20 minute walking distance of the Myeongdong Night Market and Namdaemun Market. Basically, anywhere in between Seoul Station and Eujiro-3 Station. You can get from either Gimpo or Incheon airports to Seoul Station on the AREX. Make sure to take the express train, not the all-stops local train, to save some time. (That said, the all-stop train is far superior for people-watching because it's mostly locals that use it.).

In the evening, just walk where-ever in that general area. It almost doesn't matter, you're going to be blown away by a hundred different things. In the morning, walk down the Cheonggyecheon.

Be aware that Google Maps is useless for walking directions in Korea... it only works for public transit.