I think paint over undercoating would eventually peel off. Originally the underside of the hood was not painted. Mine is black primer with some Omaha Orange overspray.
But if you want to paint the underside of the hood that's OK too. I would pressure wash it to get loose stuff off and then shoot a coat of some sort of primer before your topcoat. Dust and/or grease will definitely need to be removed. I don't think underhood heat will necessarily be a problem.
It may last long enough for your needs, but expect the paint to come off in spots.
If you're just trying to pretty it up a bit and don't need the body color, I'd just shoot a good quality undercoating over what's already there. It's rubberized and will probably stick better than paint.


Kevin
1951 Chevy 3100 work truck
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1929 Ford pickup restored from the ground up. | 1929 Ford Special Coupe (First car)
Busting rust since the mid-60's
If you're smart enough to take it apart, you darn well better be smart enough to put it back together.