How to hang a painting on plywood on the wall?
If you have some power tools, you can make a very thin French cleat.  The image below shows one designed to hold a heavy cabinet.  The two mating pieces are cut at an angle, typically 45°, and the piece on the wall captures the piece on the back of what's to be mounted.  The item just slides on top and is held by gravity (easy on and off; no tough aligning of anchors or capturing wire).  


Image by Tenbergen, courtesy Wikipedia

Your painting isn't very heavy, and the cleat would be long (you can make it almost full length, so it looks like the painting is floating a tiny distance from the wall).  It would be attached to the wall at many spots, so the load will be very spread out.  The cleat attached to the painting can be held with glue; it's a large surface area and not much weight. 

Even 1/4" material would work for your painting if you have the means to work with something that thin; 3/8" material might be easier to work with.

The cleat goes near the top of the painting.  Glue a narrow strip of the same material near the bottom of the painting, or a few scraps near the bottom corners, to act as a spacer so the painting hangs a uniform distance from the wall.