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Take the subway. Download an app called MetroidHD (https://play.google.com/store/apps/details?id=com.whitecrow.metroid&hl=en ), select Incheon Airport then Myeongdong. Follow the path. It is easy as pie and all stations have english instructions. Id also suggest going to Itaewon if you are english speaking. Has tons of clubs and eateries as well. From Incheon airport follow the path to the subway by turning right out of the gates and follow the icons to the subway (youll need to stop at the tickets booth and buy a subway pass, ~15,000Won will be enough). Will take you about an hour to either Myeongdong or Itaewon each way.
/edit six hours is plenty to enjoy either location.
One of my favorite places https://en.wikipedia.org/wiki/Gwangjang_Market. Find the bindaetteok alley (https://en.wikipedia.org/wiki/Bindaetteok) find this shop 순희네 빈대떡 (http://m.blog.naver.com/teemoticon/220699587500) it is the best. Also, Tim Burton was there once.
Public transportation is pretty easy for foreigners. The subway is easiest. Lots of english signs and info available. Buses are a little harder because not as much english signage, but the stops are announced in english on most lines. Not so true if you go to smaller cities. Get a T-Money card (https://english.visitkorea.or.kr/enu/TRP/TP_ENG_8_1_1.jsp) load it up with money in a subway station and you can go anywhere. It is accepted at all subways, buses, most taxis (unless they are sketchy) and you can even use the card to make purchases at a lot of convenience stores.
This app will let you check out subway routes, times, and other info in English https://play.google.com/store/apps/details?id=com.whitecrow.metroid
If you can read Korean, Kakaomap is the way to go https://play.google.com/store/apps/details?id=net.daum.android.map
And if you can read Korean there are some other apps for booking buses and trains to other cities. My Korean isn't great, but I make use of a lot of these. Especially Kakoamap, because I can use it for door to door directions.