Need the closest hostel possible to Frank Main Airport

I don't know of any major international airports with a "hostel" next to the terminal. Frankfurt airport has a couple of hotels connected to Terminal 1. Might be pricey, but maybe its time to open up the wallet and let the butterflies out. Or a nearby hotel with shuttle service.
By "10-15 mins away", I would assume they are located in the city center, which is indeed 10 to 15 minutes away on the S-Bahn. If low priced overnight lodging is what you want, that is where I'd look.

I don't know of any "hostel" at the airport. But it's a 11 min. train ride only into town, where you will find hostels, e.g. an A&O just a few minutes from the Hauptbahnhof.

I wouldn't bother with any of the cheap hotels near the airport that have shuttles. You will wait longer on those than you would on the train that takes you into the city and back.
Only alternative is Hilton Garden Inn which offers rooms at a decent price. It is located at the airport itself, just a 10 min. walk to the Squaire.

The Frankfurt Hostel is right across the street from the main train station. That would be your best bet. The trains run every 15 min. and it will take you 11 min. to get there. Price is 4.65€ one way. The trains start running at 04:15.