Do I really need to wait a month before applying primer to plaster?
The reason for waiting is to ensure that the plaster dries completely. If you paint over damp plaster the moisture has to still come out - through the paint causing it to bubble and come away. You will end up having to repaint at the very least.

If you have only patched the wall then you should be able to prime and paint it sooner as there is less area to dry.

However, I would leave it as long as possible before painting, just to be sure.