I made a similar short trip to Seoul last year (27 hrs) in winter - enroute to Vietnam. I stayed at the Vabien II Hotel ($84. I believe). It worked out very well because the airport bus stops right in front of this hotel, and the hotel was quite nice (a business hotel) and well located in the city hall area. I had made arrangements for the free Goodwill Guide service, and this was excellent. You really need help in Seoul if you don't speak Korean! My guide was a Korean grandpa who had worked as an English teacher.

The Vabien II is a 3* modern hotel, and all rooms come with kitchenette. I would return to this hotel - I doubt if you could find anything more convenient downtown.