Wet the buckled veneer panels with wood moisturizer. Apply multiple coats to the affected areas with a brush. Continue adding coats until the moisturizer has completely permeated the wood and the area has begun to unbuckle and soften.
Load up the syringe with wood glue. Inject the glue beneath the affected veneer. Start from the inside of the damage and work your way out, lifting the edges carefully and layering the glue underneath.
Clamp the veneer down tight to allow the glue to set. If there are no edges on which you can fit a clamp, lay some heavy objects on top of the wood -- such as stacks of books, canned items or bricks.
Close any gaps between the veneer panels with wood putty. Smooth it down with your hand to make sure the area fills up completely. Gently wipe away any excess that squeezes out, and allow it to dry completely.
Sand down the area with the sanding sponge, to smooth away any seams in the flat surface that may have been missed when you wiped the wet putty with your cloth.
Coat the area with primer, if you plan to paint the repaired veneer. If you want to keep the wood patina as it looks, seal it with a tung oil finish. Most tung oil sealants can be blended with a thinner or used at full strength. The thinner helps the oil permeate the wood fully and cuts down on drying time. It will require multiple coats to fully seal the wood. Allow 30 minutes before wiping any excess oil that is dripping from the surface. Allow 48 hours for the sealant to dry completely.