First, I’d take the Airport Limousine (actually just a bus with nice seats) to Jong-No (종노), the information desk at the airport will tell you exactly where to catch it, and then take a taxi to Gyung Bok Gung (경복궁), the largest and most impressive of Seoul’s palaces, open until sunset (not everyone finds their way to the far end of the palace, but it’s my favorite part).
After an hour or two of wandering, I’d head out the west-side gate, near the folk museum and large pagoda-like building, cross the street, turn left, and explore the restaurants and coffee shops of Samcheon-dong (the European restaurants are good here, but not really the best place for Korean food). If my legs aren’t too wobbly from the flight and the palace, I’d climb the very steep steps and keep going a few blocks into Bukcheon-dong Hanok-maeul (Hanok=traditional house, maeul=village), a beautiful area of persevered and restored aristocratic houses from the 18th century.
When I’ve had my fill there, I’d flag the first taxi back toward the city and head to Jogyesa (sounds like Joe-gay-sa) the head Temple of the Jogye order. Don’t be shy to go sit, or at least walk through the huge main hall, just take off your shoes first…
The next street over (across from the temple) is Insa-dong, a tourist street filled with traditional shops, tea houses, restaurants, and street vendors. A lot of stuff is a bit over priced, but you can find some nice stuff, or just enjoy looking. From here, you can follow the street back to where the bus first dropped you off. If time/energy allow, across the street is Seoul’s biggest concentration of neon-lights, lots of restaurants and bars. Just behind it is Cheongyecheon stream, a nice place to sit, deep your feet in the water and relax before catching the bus back to the airport.
These are my suggestions, they’re all relatively close together, the palace is just over a 5 minute walk from the temple, the traditional village about 15, but a steep 15, that would be the most energy consuming. If you do head up the hill, there’s a beautiful traditional tea-house that’s worth checking out, too. Taking the cable car up to N-Seoul Tower is fun, too, really great views of the entire city. It’s nice after dark, and not far from Jong-No. Also, the tea shop that I’ve spent many hours in over the past few years is just a minutes walk down from the temple. They’ll invite you to sit and drink tea with them for free, let them know I sent you, and they’ll be happy! They can also point you to a really good traditional restaurant just around the corner where you can get some really good bibimbab, among other things. Careful for the “yuk hwae bibimbab” it has shredded raw beef. It’s not bad, but you probably don’t want to end up with it anyway! ^^