Happy New Year to you too :)
If you want to travel via plane I'd highly recommend to have a proof of identity (passport/ID) with you. Airline staff quite often doesn't check your documents for flights within Schengen or domestic flights (aka flights within Germany), still you need to have a passport/ID with you in case they want to see it. Otherwise you might not be allowed to enter the plane. Your Aufenthaltsgestattung alone is not sufficient to prove your identity.
So: Yes, you can buy tickets and try to get on the plane, but you might experience a rude awakening if they ask for a proof of identity other than your Duldung :/
All the best,