you will be so tired you won't know the difference between the Hilton and the third chair on the right in the terminal. Mongolia is beautiful enjoy.

The best place to stay… that includes all of the things that you want is… A JJim Jil Bang.
Wont cost you anymore than $10US a night… but there is no bed, bet there is nothing wrong with sleeping on the floor. Some places have sleeping rooms, where you will get a mat and a pillow.
Its not private though… but we often go there for the night.
They have gyms, restaruants, movie veiwing rooms, many many fantastic sauna rooms, public bathing rooms with many different pools to choose from.
Massage, Hair salons, PC rooms… everything under one roof.

Not too sure how far from the airport you are wanting to go.
Incheon airport is quite far from Seoul city (about 1-2-3 hours depending on traffic) however I have heard of fantastic JJim Jil Bang in Incheon city, which is closer to the airport.

Enjoy, and as its only for the night, you might as well leave your luggage at the airport. You dont need anything when you get there because they give you some shorts and a teeshirt, and you can buy everything else inside.

People wont think you're strange for sleeping there, people go there as families and sometimes spend the whole weekend there.

Have Fun!