Airport hotels are expensive. There is only one sleeping pod available AFAIK it is the Yotel terminal 2E, airside, you need a boarding pass. Check the terminals with flight numbers.

Hotels landside are around 100€.

Sleeping inside the terminals landside or airside is possible as long as you have a boarding pass/reservation. Best option is Terminal 2E (more seating option, cleaner environnement)

Innside by Melia is a new hotel, decent, and not overpriced. Moxy also is a good option, both available landisde for under 100€.

Nothing to worry about going through passport control to sleep in a hotel airside, especially that early in the morning, but you have to decide if it's worth it for just a few hours sleeping.