Even if your neighbour is responsible for/owns the fence that doesn't necessarily mean he must repair it. He can choose to do without it if he wants and has no livestock to keep in. If it's unsafe he must make it safe, by removing it if necessary, but that's all.
Unless his deeds dictate that he must keep it in good repair, he has no obligation to do so. If they are the same as yours it's likely that ownership and responsibility are undefined.
So sound him out gently. If he agrees it's his responsibility, you are in luck. If he says it's your responsibility, you are out of luck. Offer to go halves.
Ultimately he who wants the fence pays.