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.
Check out the Berlin Airport Public Transport Page
S-Bahn Trains are your best bet (these run every 24 hours at 30 min intervals at night on weekends)
S9 every 20 minutes (Pankow – Schönhauser Allee – Ostkreuz – Schöneweide – Adlershof – Flughafen Schönefeld)
S45 every 20 minutes (Südkreuz – Neukölln – Baumschulenweg – Schöneweide – Adlershof – ...
The Kubrat Hotel (Checkpoint Charlie) at the Leipziger Straße 21:
You use the Airport Express RE7 or RB14 in direction Hauptbahnhof which is coming each hour. You leave the train at Friedrichstraße after 20-25 min and enter the U6 in direction Alt-Mariendorf. After two stations you leave the U6 at Stadtmitte. You go in direction Leipziger Straße, after 60m ...
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.
An email to the airport yielded the following answer:
Thank you for your enquiry.
We are glad to inform you that you may use any preferred security line
(in any terminal) to enter the security area at Schoenefeld Airport.
Further information you may acquire via the following link:
Some S-Bahn/U-Bahn lines operate all night long, at least on some days, and there is also a network of night buses. Otherwise, the regular S-Bahn service starts at 04:30 so depending on where you are staying the first S-Bahn (perhaps combined with a bus) could be the solution.
In addition to all the resources mentioned by EdmundYeung99, you can simply use ...