It's really not a big deal. Don't panic. I saw no need to checkin at the Hyatt during my overnight there last July. We just rode the shuttle bus back to the terminal and checked in as usual at the Korean Air Prestige Class desk around 7:45 am for our 9:30 am to ATL. We had no wait or a minimal wait at checkin. Security took about 15 minutes to clear; no priority line, but Incheon knows it's peak time so they have a good number of lanes open. Outbound passport control took a few minutes, no more.

Situation may be a bit different checking in for Economy, but as far as I could see, there were plenty of checkin desks staffed and the lines were moving.

Inbound customs clearance is no big deal. Only snag inbound likely to be passport control for foreigners. I'd suggest hustling so you get there before the rest of your flight and hope an A380 didn't arrive right before you.