I assume there is a way to export stills at other than screen resolution.
There isn't. Screen resolution has little connection to print resolution. Video (and film) doesn't have any sort of DPI, though it's commonly referred to as having 72DPI; this is really just a convenience, more than anything. The only thing that matters in video is raw frame size, e.g. 1920x1080. At the typical print resolution of 300DPI, that's about 6.5"x3.5"--not exactly poster size.
The best you can do is use Photoshop or other scaling algorithms to blow up the image, with the knowledge that it's going to get less sharp the larger you go.
You can't - I think 72dpi is all you get
Video is based on the size of the codec, not on pixels
DPI is completely irrelevant in video work. Exporting a frame from PR is only about pixel dimensions, for instance 1920 x 1080 or 720 x 480. There is no relationship to DPI. For printed work you may need 300 DPI or even higher, but you will be left with a lack of pixels to make a decent print. If the pixels are missing to start with, there is no way to create them.
There are actually several software programs that promise excellent scale-up for exactly the conditions you specify. Just google; caveat emptor, of course.