Depends on the area of the bond. Gel CA might work but would be expensive for a large area. Epoxy would definitely work but also expensive. If you are careful to smooth it out uniformly, construction glue (like from a caulking gun) will work.
If it is a large area, I would prime the plywood with primer and then use spray can contact cement, spraying both the primed plywood and the styrene. This will be an instant bond, so you must carefully position the styrene before the surfaces touch.
You might consider a wood plaque from a craft store, like Michaels, instead of plywood. Smoother surface will bond better than plywood and the edges are already routed with a decorative edge.