The grease imbedded in the finish is what could be a future problem. If
you can really clean it good then prime and paint, Some old cabinets
are well kept up, some the finish is junk. If the old finish is ok and
doesnt come off with your nails then ok paint if its soft and removes
easily especialy around handles and knobs it has to be stripped . Over
many years grease and oil usualy penetrate finishes ruining them,
every job is different.