It's got chicken wire on it... you'll likely want to replace it with hardware cloth. When considering how much it costs to build something, don't forget to assign a value to your time and effort and stress. Even for me, knowing how to build stuff and having the tools, at $120 I'd just buy the darn thing and replace the wire rather than go to Home Depot, load up the lumber, drag it all out in the yard, etc etc etc etc. And, if you want to build another one, that one looks like it would make a good pattern to work from.
In Louisiana, enclosing the walls with boards or bits of tin will likely be sufficient for winter. Right now, with the humidity and heat index, I'd leave it wide open.
Edited to add: I don't see any diagonal bracing on it. You might give it a good shake while you're there, and offer a lower price if it's wiggly, then get it home and screw a 1x4 across each end from the top of the short wall to the bottom of the tall wall, and a couple across the back.