Some neighborhoods that are active at night would be Gangnam, especially around Gangnam station. Hongdae and Itaewon are also lively at night. All of them will have some good places to eat and good hotels nearby. All of them will also have airport limo bus stops in the area.

For food, the area between Nonhyeon station exit #2 and Shinnonhyeon station exit #3, there is an alley parallel to the main street that has a bunch of restaurants that stay open late. This is in the Gangnam area.

You will have plenty of time to sleep on your flight to NYC so maximize your layover in Seoul and enjoy it.