I think here's what you can do to enjoy Seoul with the limited time you have.
Ride the Airport Limousine Bus No. 6011 from the airport to Anguk station, costs 10000W and takes about 1 hour.
Enjoy the very interesting Insa-dong weekend market next to Anguk station for 2 hours, then just cross the road and explore The Gyeongbog-Gung Palace for 2 hours. Or if you like you can just spend 1 hr around Insa-dong weekend market, 1 hr around The Bukchon Hanok Village and 2 hours for The Gyeongbog-Gung Palace, these are some of the major tourist attractions in Seoul and they are within walking distance of each other.
Head back to the airport 2 hours before departure on the same bus .
That's it.
PS. in regards to the weather, I am pretty sure it is a lot colder in Dec as when I was in Seoul near the end of Oct 2013 it was like freezing at night.