I'm ripping off my nasty carpet and talked to my contractor friend and she said to paint the plywood with a water based opaque deck stain. She mentioned the brand Fortune. Then you seal it with a gloss or whatever finish. Do not wet the plywood, it needs to be cleaned without getting it wet. I'll let you know how it works.
The cheapest paint-grade plywood would be BC grade. It's smooth finished on one side only. If you need something nicer, you can go with any smooth plywood. Check for divots, or "footballs" (the plugs they use to fix divots found at the plant).