How to close a gap between the fence and the neighbor’s that’s on his side of the property line

Since you just want to get the fence back up: screw a long board horizontally across the pieces of fence, bridging the broken post. A couple of long 2x4's should do the trick.

Skip fixing the post altogether.

Your iron fence, like this:

... Is the more traditional choice. A fence like this is going to be very durable, generally only requiring a coat of oil paint every few years and replacing minor touches like the caps on the posts. These are also generally more aesthetically pleasing as they are typically installed as part of the overall decor of the pool. However, iron fences are generally more expensive, especially as a retrofit; the normal installation method is to anchor the posts in concrete, which would require drilling through your existing concrete pad to set them. It's also possible and accepted to "surface-mount" this type of fence onto an existing pool deck using L-brackets welded to the posts and lag screws for wood, or concrete anchors for a pad. This isn't as strong, as the screws can pull out over time, but it's usually more than enough. Lastly, an iron fence hurts more to run into; they don't have much give (not any if properly installed, really).

A vinyl mesh fence, like this:

is a newer option. These generally cost less, are easier to install, and the aesthetic can be pleasing (instead of the thick iron bars, you have a filmy mesh, reminiscent of a privacy screen). Along those same lines, the mesh provides slightly less visibility than the iron, especially in low light. This can be a two-edged sword; the neighbors can't see you skinny-dip, but you may not see a child falling in the pool in the evening or at night. Before you go this route, ask the supplier for a demo panel you can set up between your pool and an indoor vantage point to check the visibility as the day progresses and make sure it's to your liking. The mesh has some give, so a young child running into it won't be injured as badly (it may still throw him back onto the ground).

On the downside, the mesh screens are more easily cut or torn, and if this happens the entire panel can rip in half allowing pets and kids into the pool area. Do not use these if you have big dogs that like to scratch on the back door; they'll do the same on these and break through pretty quickly. The mesh is replaceable, but it's a hassle and a safety concern. These fences generally come prefabricated for ease of installation, but that generally means there's one way to install them; anchored with L-brackets to your deck. An adult running full-tilt into a panel could break the fence down. And while they can look good, they can also look cheap (usually due to poor installation, or simply because a particular mesh fence product IS cheap).