12 hour layover starting at 4am

Hello,

Traveling through, and the person I'm traveling with and I are at Incheon for 12 hours, but the layover starting so early makes a lot of the tour schedules not line up so well.

If you had to venture on your own starting really early in the morning, where would you go, and what would you do, to Be able to hear back to the airport and be there by 2?

Looking for anything interesting to see or fun to eat.

Advice appreciated.