Take the light rail/subway from the Incheon area into Seoul proper if you can, not much to see or do by Incheon. About an hour each way, but worth the trip.
For a night, I recommend heading towards Hong Dae area (Hong Ik University). Fun to walk the street and see the performers, lots of energy and fun vibe. Good restaurants all around, but my recommendation is Ichiryu. My uncle showed me this spot, very good lamb bbq in a somewhat traditional manner and a unique experience. If you want a more familiar “Western” experience, head towards Itaewon. As an American I felt comfortable there. Head towards the main alleys that run parallel to the big street on either side for food and fun.
For the next morning, if you did the proper Korean thing and drank too much (soju) the night before, you may want hangover soup. Sun jee hae jang guk (cooked blood hangover soup) at Cheongjinok (청진옥) in Jongno is an institution. For SamGaeTang (whole chicken soup) i recommend Tosokchon near
GyeongBok Goong. I go every time I find myself in Seoul.
For lunch/street food experience, GwangJang Market is unbeatable (see Netflix Street Food). The bin dae ttuk (mung bean pancake) is amazing, and soon dae (blood sausage) is delicious. For Kal Guksoo (knife noodles), my favorite is Myeong Dong Kyo Ja in Myeong Dong market. Great dumplings there too.
