How can I prepare a stage backdrop that serves as a projector screen at the same time?

That might not be a great idea because the people in front of the screen will cause shadows on the screen, but if you want an inexpensive way to do it, you could get several wooden palates stacked on top of each other with butcher paper spread out and stapled onto it.

Not necessarily Buckyboy28, you could rear project the images then you wouldn’t have shadows. Also you can front project without hitting the people in front with the right angle with a projector hung from overhead. It is best then to have a projector that you can do corrective skewing to compensate for the weird angle. Check out Rosebrand online. They are a theatrical soft goods company and they have a lot of experience with projection materials.