Sure, you can potentially paint on any fabric. As far as the priming, you can use gesso or PVA glue, which is also archival. There is also the option of priming with a transparent acrylic medium, like a soft gel, that would leave the fabric texture and design completely viewable.
For some fabrics that may be too thin to stretch, you can glue them to a hard support, like a wood or MDF panel, using PVA glue or acrylic medium.
Keep in mind that you can technically paint with Acrylics on unprimed fabric and no hard will be dome to it. Oil paint, in the long run, can rot the canvas, but acrylics are fine. If the fabric is thin, it does not soak up all the paint - which makes your job harder - and especially if you like the pattern design, you may not want to cover it with gesso.
A word of caution for upholstery fabric: often it's treated to be stain resistant and water repelling, I'm pretty sure that would affect the priming and painting. Prewashing might help, but I'm not sure.