Every problem in an AnCap society is an opportunity for someone to find a solution to that problem. And even every solution is an opportunity for someone to come up with a cheaper solution. If you think an AnCap society wouldn't be able to organize to expend real resources on preparation for a threat that might not materialize, I would just point out that no known society is immune to that problem. But the general solution is to get as rich, prosperous, and technologically advanced as possible and hope that will make any problems, expected or not, easier to solve.

If you look at the relationship of governments to each other in our world, it's somewhat like an AnCap society. There isn't really any "super government" that rules them. And the UN and similar organizations are voluntary associations among those countries that can be abandoned by their members at any time. There isn't really any socially-accepted world policeman with any kind of monopoly and countries use both their economic and military might, both through threats and actions, to 'persuade' each other. The analogy isn't perfect, but it does give you an idea of how these kinds of things might go both wrong and right in an AnCap society.