Haven't stayed at the Sheraton but have spent many nights at the W right next door. I would try to see if you can get into the W Walkerhill as it is a fabulous property (get a media room!!). Anyway, the Walkerhill area is a significant distance from the airport, likely an hour in a taxi and is fairly remote (I used the W there as a get away from the bustle of Seoul). If you want to see the city you may want to try going across the river to Coex mall which has excellent shopping and people watching. If you want a much longer trip you could head to Itaewon (famous for clothes) or Namsam park.
J