Would this mean that Agnew would become President upon Nixon’s resignation?
That's how the constitutionally-dictated presidential line of succession works, yes.
If so, what would President Agnew be like?
An oaf without astute political skills who just delegates his foreign policy to Kissinger & domestic policy to advisers who actually have brains.
Would Reagan primary Agnew like he primaried Ford?
If he senses blood in the water (i.e., a ripe enough opportunity to potentially win) as he did in real life, then I see no reason why he wouldn't.
