I'll be on a comparable trip in two weeks.
I've opted for this:
- store luggage at the airport ($8 per day)
- take AREX into Seoul station, sightseeing
- take AREX back to the airport in the evening
- pick up luggage, head to airport hotel
- sleep long, head to airport next 'morning'
The benefit of Incheon is that there are plenty of cheap hotels with airport shuttle busses available.