I just checked a map and there's the 신공항모텔 Shin Gong Hang Motel. The number is 032-751-8818. It says Day Rental is 30,000₩. Doesn't list the hours, but it's usually three or four hours. Just have the hotel call them when you land. You can't book day rooms at motels in advance. It's 20 minutes by train from the airport, just two stops on the Arex Local.