Assuming it's standard 1/42" veneer, the best way to get it off is to sand it off with coarse sandpaper (in a powered sander, belt or ROS).
The best way to get new veneer is to buy commercial veneer. Woodworking stores, such as Woodcraft and Rockler, used to sell small bundles of veneer. Only problem is that you may only find fancy maple, not plain maple.
Regular PVA glue will work fine for this but think about how you're going to press it. If the drawer is flat, you can use waxed paper and a caul, clamped with standard clamps.
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