I think. Is the rule that one may not be, or that one may not stand as a candidate for, the President, if one is foreign-born (Arnie) or has previously served two terms (Bill)...? I check….and:

Almost nothing about the VP, and nothing relevant about qualifications (hence Cheney's ability ot argue that the VP both is and isn't part of the executive branch, I guess).

Arnie misses out:
Article. II. - The Executive Branch
Section 1 - The President
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.

"Not eligible": doesn't matter how he'd get there, whether by vote or succession as VP.

Bill however...:
Amendment 22 - Presidential Term Limits. Ratified 2/27/1951.
1. No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. …

Not to "be elected" more than twice. To become President a third time, in a manner other than election -- no rule against that.

So -- yay Bill! Only, he can't do it as First Husband, because not only is Hillary unelectable, but electing a married couple would be too weird even for the US...surely?