There are various possibilities. Not Kennedy - he was driven by ambition, not agenda. Not Johnson - this is an AH, and Johnson liked being President: he too was ambitious.
It has to be somebody who didn't seek the Presidency, didn't want it.
Timing is also key. Ike was 34th. This man is 36th. He has to be VP to Ike's successor; he could succeed to the Presidency either in 1961-1965 or 1965-1969 (if Ike's successor is re-elected in 1964). Then he has to live till late 1984 or 1988. (It seems unlikely that an "accidental" President would be re-elected in 1968, so not later than that.)
Actually, there is one other possibility - Ike's successor (35) serves two terms, but the winner of the 1968 election dies before taking office. So 36 is "accidental" and serves a full term; the interview with Ambrose could be in 1992. (Or the presidential candidate dies just before the election and the VP candidate steps up and wins anyway, due to circumstances,but is accidental.)
Or ultimate weirdness: a Wallaceoid Dixiecrat throws the election into the House, which deadlocks. But the Senate elects a VP: the party with a Senate majority elects their candidate who ran second, for party reasons and because the leading VP candidate has become viewed as nasty (he became a drag on the ticket, even his own party doesn't like him). Whereas the second-place VP candidate is a "statesman", with, initially, a better reputation than either P candidate.