Baring major reforms the NCR could last about 10 years if they don't handle the famine situation, though I imagine they would. In any case at min. because of the institutions backed by written laws they're stronger than the Legion and it's random succedion problems.
Famine wouldn't end the NCR. Famine hasn't ever
ended a country. Look at Ireland, it underwent severe famine, and yet it's still around today. Some of the population dies off or leaves, and then there's enough food again. Or the government finds a way to produce more food.
EDIT: Of course, this would considerably weaken the NCR, and with a combination of outside forces, it
might be able to bring it down, but by itself, no.