Some tips for painting veneer cabinets include cleaning the surface using a trisodium phosphate product and bonding any raised part of the veneer before starting work. Some other tips include using a bonding primer instead of a latex primer and using oil-based gloss or semi-gloss paint.
Before painting, take the cabinet doors off of the frame, and remove all the hardware from the doors. Use masking tape on the back of the doors to mark where on the frame the door belongs, and place each door's hardware in separate sandwich bags marked for its corresponding door. Use trisodium phosphate cleaner to avoid dissolving the veneer glue because of harsh cleaning products.
Use a cotton swab of wood glue to work glue into section of the veneer that have lifted over time. Allow the glue to dry while holding the veneer against the base of the door. Use painter's tape to cover any area of the cabinets, trim or wall surface that you want to protect.
Use a paintbrush or medium-nap paint roller to apply the bonding primer to the surface of the cabinets. Once dry, use a sponge brush or light-nap roller to apply the first thin coat of an oil-based paint. Allow the first coat to dry, and add another thin coat. Repeat until the cabinet is as opaque as you desire. Allow the cabinets to completely dry for about 48 hours before attaching the hardware and hanging the doors back on the frame.