The original fence is wood and could possibly be repaired. The neighbor is putting in a pool and wants to install a better fence.
Real Estate Attorney
Your neighbor can't force you to pay half the cost of a better fence. It sounds like the existing fence may need to be replaced. You could offer to pay half of what it would cost to replace the existing fence. There are potentially serious liability issues associated with a swimming pool which could come into play if a child or disabled person gained access to your neighbor's yard. You should discuss the issue with your neighbor to see if you can reach a mutually agreeable compromise.
This answer does not constitue legal advice, nor does it creat an attorney/client relationship. If you are seeking legal advice upon which you intend to rely, you should hire competent cousel familiar with this area of the law in your locale.
Construction / Development Lawyer
The general rule is that each neighbor is responsible for one half of the fence.
But, you should not be penalized because your neighbor wants to put in a more expensive fence or because he chooses to put in a pool. A wood fence is perfectly legal for a backyard that has a pool.
If the current fence can be repaired, then tell your neighbor you are happy to pay 1/2 the price for the repair. If he wants to put something else in, then he should be responsible for anything above one-half the repair costs.
The neighbor can't force you to pay, but he could file a small claims case. Then it would be up to a judge to decide. Take pictures of the current fence and get an estimate for the repair, just in case.