I think Reagan might have run in 1984, but skip 1980. 1984 would be absolutely the last chance for Reagan to run for a first term as President though.
It might have looked like the following:
Gerald Ford (1974-1981)
Jimmy Carter (1981-1985)
Ronald Reagan (1985-1993)
Mario Cuomo (1993-2001)
Bill Clinton (2001-2005)
Rudy Giuliani (2005-2013)
This is very likely, though I only see Reagan serving one term under this scenario.
I agree, Reagan would probably step down. My guess is that Howard Baker or Bob Dole would be the nominee and successor
