This is what my "crazy" sister made for my sister-in-law who is laying over in Seoul, Korea for 10 hours. My sister-in-law decided to go with the option 2 and she thoroughly enjoyed lay over in South Korea!

Two Options Recommended

1) If you are tired, stay in the airport. You can still have fun.
2) If you want to go out and experience both local and tourist areas + local shopping, take the option 2.

Option 1: If you are tired, stay in the airport

1. Spa on Air
Since you are tired from the flight, you can first rest at Spa on Air (B1, East side) It costs 15000 won (about $15) and extra 10,000 won ($10) for private sleeping room.
http://www.airport.kr/airport/facility/efalicityOpen.iia?carId=39
 

2. Internet
There is free wifi available but if you are having trouble with connection, there are several internet lounges that you can check emails and browse internet.
Free: (Duty-free Area) Concourse 3F near Gate 111, 124
With a fee: (Duty-free Area) 3F near gate 14, 30 / (Public Area) 2F Near Arrival A, E
http://www.airport.kr/airport/facility/efalicityOpen.iia?carId=30&facilityId=347

3. Food
Restaurants (3F) operated by hotel or big corporations. About $15-$20 per meal
http://www.airport.kr/airport/facility/efalicityInfo.iia?carId=41&facilityId=443

Food courts located in B1, East side (usually for flight crews and airport employees). About $5 - $10 per meal
http://www.airport.kr/airport/facility/efalicityInfo.iia?carId=14&facilityId=406

Both locations offer good quality of food and various choices. You can walk around and look for what you like. If you want my recommendations, I have been to “Kayakeum”(restaurant) and “Food on Air Studio 2” (food court) and both of them were good.

4. Starbucks
There are many coffee shops at the airport but if you need to find Starbucks, there are 3 Starbucks locations at the airport.
http://www.airport.kr/airport/facility/efalicityInfo.iia?carId=17&facilityId=397
(Duty-free Area) 3F, Near Gate 30
(Duty-free Area) 3F, Near Gate 28
(Public Area) B1, East side of Center

5. Korean culture experience
There are museum, cultural experience zones, and gallery at the airport where you can experience traditional Korean culture at the airport.
http://www.airport.kr/iiacms/pageWork.iia?_scode=C2605010100

6. Duty free shopping
Duty free shops in Korea are very big and have the latest, trendy, luxury products and goods. The pricing is pretty good and selection is excellent because a lot of Koreans shop a lot if you are looking for non-USA luxury goods. Korean girls spend 2-3 hours for shopping before their flight. For the local Korean products, it tends to be expensive and the choice is limited than local stores because it is tourist area.

Option 2: If you want to go out and experience both local and tourist area and local shopping

I recommend you to visit Gwanghwamun district because this is where you can experience both local and touristy in a short time. You can first visit Gyeongbokgung, Korea’s beautiful royal palace built by Joseon Dynasty in 13C, and then you can visit Seoul KTX (Korean Train eXpress) station and have lunch at the food court or many restaurants at Seoul KTX station, and do shopping at local mart “Lotte Mart” (where you can try and buy Korean snacks, food, cosmetics, accessories, bakery and etc. at regular local pricing) and the department store “Concos”.
If you shop at the airport, most of them are duty free shops and tourist shops so it could be expensive. But “Lotte Mart” at Seoul KTX station is a regular local mart that local Korean people go to and is a popular / recommended shopping place for foreigners at the same time because the location is connected to KTX train station and has easy access from/to the airport via airport train.

Take the airport limousine bus from the airport to Gyengbokgung so that you can do sightseeing of Seoul on the way there, and take the airport train back from Seoul KTX station to the airport so that you can be on time without traffic (traffic can be bad in the afternoon).
If you are thinking of getting tour guide at Gyeongbokgung, the first English session starts at 11:00 am so have breakfast at the airport or rest a bit at Spa on Air and take the limousine bus around 9 am.
If not, you can go to Gyeongbokgung as soon as you arrive and finish your tour early and come back early so that you can spend relaxing time at the airport before your departure.

1. How to get to Gyeongbokgung

Go to the exits 5B or 12A on the first floor and take bus #6011 bound to Seongbook / Wolgye operated by Airport Limousine. The ticket is 10,000 won (about $10).
The stops are: World Cup Stadium, Seongsan Hall, Yonsei University, Ewha Womans Univ Back Gate, Gyeongbokgoong stn, Angook-Dong, Hyehwa Station(Sungkyunkwan University), Hanseong Univ. Station, Seongshin Women's Univ. Station, Miari-gogae, Gireum New Town, Dream Forest, Lotte Castle Lural Apt, Wolgye Station(Induk Institute of Technology)
The bus runs every 20-30 minutes.

http://www.airport.kr/airport/traffic/bus/busList.iia?flag=E&fake=1306823313914

It takes about 50 minutes and you need to get off in front of Seoul Police Agency near Gyeongbokgung station.
Use underpath from Gyeongbokgung station (exit 7) and come out to exit 5. Exit 5 is connected to Gyeongbokgung.

2. Gyeongbokgung (Royal Palace)
Admission fee is 3,000 won (about $3), and tour starts in front of Gyeongbokgung reception office inside of Heungnyemun Gate. Tour lasts for an hour to an hour and thirty minutes.

http://www.royalpalace.go.kr/html/eng/guide/guide01_tab01.jsp?dep1=1&dep2=1
Tour in foreign languages
English: 11:00, 13:30, 15:30

3. From Gyeongbokgung to Seoul KTX station
Take a taxi from Gyeongbokgung and it will cost less than 5,000 won ($5) and it will take about 5-10 minutes. Tipping is not required nor expected in Korea.

4. Seoul KTX Station, Concos Department Store, Lotte Mart
There are restaurants and food courts in Seoul Station as well as in the mall connected to Seoul Station.

5. From Seoul Station to Incheon airport

There are two types of trains: express and train for commuter.
With Express, it costs 13,800 won (about $13) and takes 43 minutes and runs every 30 minutes. You can check-in and do immigration there if you take express because it is airport terminal.
With regular train, it costs 3,850 won (about $4) and takes about 55 minutes and runs every 10-15 minutes.
You can take taxi (costs about $70 - $100) instead of train but it is recommended to take airport railroad on the way back to be safe with time.

http://english.arex.or.kr/jsp/eng/index.jsp