"Shot a coat of primer" is correct. Painting cars is done with a spray gun, so you would 'shoot' or 'spray' a coat of primer on the auto body. 'Shot' is just past tense.

All in all it means that his daddy bought an ex-police car (already high performance), sprayed over the logo with just primer, and then turned their attention to the engine, tearing down a high mileage engine for a re-ring, maybe boring out the block, or other preformance modifications.

In fact, NASCAR racing has its roots from bootleggers who did performance modifications to their cars.

Also, it's "big block Dodge" not "big black Dodge". This refers to they type of engine, a "big block" engine.