Property Fence is Beyond Repair - Who's Responsible for Replacing the Fence?

Look through your chain of title. Often, when neighbors install a fence, they also agree to a covenant, binding on successive owners, that maintenance costs will be divided evenly between the property owners. If such a covenant exists, then you can enforce it. But, absent some sort of agreement, the neighbor generally has no duty to provide any maintenance at all, so you should happily accept the $500, which is better than $0.