I fly into Frankfurt at 9pm, and fly out the next morning at 9am.

I am on a budget so I don't want to book the hotels inside the airport (expensive). I saw a cheap hotel nearby the airport (but the airport is huge, so I imagine it would still take a while to get to it).

I'm guessing I'd get to the hotel by like 11pm, be in bed by 11-12pm, then have to wake up at 5am to be at airport at 6am.

Was thinking of the "Meininger" hotel, so I'm guessing I'd have to take a Uber to and from there (extra $$).

Would it be better to just sleep in the airport, if I have that little time?

(The thing about that is I'm flying on 2 separate tickets (flying to FRA from within the Schengen zone, then catching an international AA flight to the US the next morning)... so I have to pickup my baggage and then recheck it the next morning before I fly out of Terminal 1. And I hear part of the airport shuts down? So not sure if I'd even have access to the terminal that's comfortable to nap in).