No it wouldn’t beause the VP has to meet the exact standards as the prez. Ineligible to run for prez, ineligible to run for vp.
Let's say, for instance, that GWB decided to run for Congress, he won, and then was elected Speaker of the House--third in line for the Presidency. And then let's further say that both the President and VP died in office, making the Speaker of the House the President. Could GWB serve? The language of the 22nd Amemdment certainly seems to support that position, since he wouldn't have been elected to the office of President more than twice.
The VP issue is admittedly a little less black-and-white, but the same principle applies. The 22nd Amendment doesn't address eligibility, just electability.