I have been to Schönefeld a few times before, but I cannot remember:
Is there an easyJet stand or an office where I could buy my ticket (in advance, I mean) using cash or a German ATM card?
I don't have a credit card so I can't buy online.
There is an EasyJet Customer Service desk where I had to go to change my flight because I had just missed the one I had originally booked. They handled everything there and printed me a new boarding pass, so I presume you could book a ticket there too. Whether they take cash or not is a different matter, so it'd be best to go there beforehand to find out.
You can find the desk on the ground floor near the EasyJet baggage check-in area. It's right in the corner, and there's usually a helpful person standing around you can ask if you can't find it - it's not immediately obvious.
I don't remember one either but it seems you can book online and pay by bank transfer instead.
It's not directly an answer to your question, but an alternative (easier!) solution to your problem might be to ask a friend or family member who does have a credit or debit card to buy it. There is no problem with an Easyjet flight being bought on a card in another person's name, I do it all the time when travelling for work.