I just got my horse about two months ago. I think the most important things would be to go see the horse and be there when its lead out of its stall, groomed, tacked up. Watch the Owner ride then ride yourself so you can get a feel for the horse.
Somethings you could ask would be
Does the horse have any medical issues, for example my horse is prone to ulcers, so I knew when I got him that would be something I would have to treat/prevent. I'd ask how long the owners have had her and if she has had any medical issues during that time.
Is there any vices? I know this one horse who absolutly hates being sprayed with fly spray. Or does the horse tie, is good in cross ties etc...
But like I mentioned it is best to go there and see everything, see how she responds to everything, don't go after she has been tacked up, watch and ride. That's the only way you could really tell.
But I would definitely get a vet check done if the visit goes well, I was lucky and my horse had just gotten leg Xrays before I came to see him so I didnt have to pay for that, but I would highly recommended it.
And im sure when you get there and see the horse in person you will have a bunch of questions pop in your head.