The rear of the cabinet could be replaced with a new piece of ply. All other cabinet areas need to have PVA glue applied (plenty of it) to hold it all together and prevent from getting any worse.(Allow to dry for at least a week).
Apply a 25mm x 25mm PVC strip to the base of the cabinet as pictured (Glue with Liquid nails and clamp), paint to match. Others may have more ideas!

Sounds fair, will be fun to match the old paint/ink
PVC strip is good idea and yours looks pretty good