I can tell you some basic things- most which you probably know already.
As other's said, find out what the lease intails- you the only rider, or being used in lessons, other people lease... my situation was I did a share board ( I was to eventually take over ownership) So, I paid a smidge over half of what it cost to keep the horse. The horse was used in lessons so I had to yeild for that ( which was usually thursdays and sat) other than that, I had the right away. ( occasionally i'd come by a thursday, and have to wait which was fine). Plus side- I leased for very cheap. Down side- anything went wrong- i'd pay half. And it did. Actually, I disagreed with the initial diagnosis as to why the horse was lame, so I called the vet out and got x-rays and paid for it myself. The horse had navicular's disease not laminitis. So, i paid the vet bill.. and I still had to pay the lease even though I couldn't ride. ( that may be another important aspect-find out if you are expected to lease if the horse goes lame, be it your fault, not your fault, or someone elses fault if the horse is still used in lessons)
I also had free use of her tack( which included the horses winter blanket on the few rare times thehorse needed it), tho I bought my own bridle for cheap and started buying my own grooming supplies, but it will be good to know if you need to buy anything. I still use some of her equipment with her permission for my own horse ( as she uses some of mine and we are ok). Now, I personally don't agree with buying your own saddle- I know some places do this for a leased horse, but I don't see the point- what are the chances of it fitting your next lease horse or horse you eventually buy? But its something I would ask.
Another thing- are you paying monthly or do you have to commit to a time period? I did month to month, but mine was a special case and usually it was 3 months payment up front.
For the horsy stuff:
The very first time I rode the horse I was leasing when I leased- I was a tad nervous just because it was the first time without my instructor ( been riding there for 3 years). No real reason for being nervous, I just feared something would go wrong because if I don't have bad luck, I have no luck. So, just take your time. Saddle up slowly, warm up slowly, heck, you can even ride slowly. I am not the best at cantering, so I just walked/trotted, but mostly walked. I really observed the horse to make sure he wasn't upset with it just being him up there ( this horse was fine riding by himself)
One time I did start trotting too soon and the horse didn't feel right, so naturally I stopped riding and called my instructor down. I had checked all his hoofs and so did she, and she said I prolly just didn't warm him up enough ( it was winter) It was embarrassing- BUT if something doesn't feel right- stop. My instructed even said at least you had the common sense to stop even if it was something so silly as I should have warmed him up a little bit longer. Then, my routine involved a lot more hand walking to ensure I didn't make the same mistake. On the other hand- during the summer, the horse ended up being lame and I was the one one noticed it-honestly, I suspected something was off just walking up to the ring, did a nice long warm up, mounted, and when I eventually trotted it was obvious.
No one wants to not notice something wrong, but when you lease and ride on your own, I think having a keener eye to pick up if something isn't right is a good thing.
If you are leasing to prepare for horse ownership- take more interest in how the horse is fed, how to check to make sure the horse isn't loosing wieght, and keep a keen eye on his hoof care. That was one thing I noticed- I worried a lot more about what I was going to do when I rode and how everything else was put together. I got a nasty hoof care 101 with this horse which lead to me not purchasing.
You don't always have to ride when you go up. I spent A LOT of time just grooming or walking the horse around the ring.
Also, the horse got scratches, so there was a month time I stopped by nearly everyday to tend to the injury. The owner did too- we planned it out, but it really gives you the sense of the responsibility of caring for a horse and helps bond with the horse.
As far as the arthritis- find what the owner suggests ( most important), get advise here, and read up on it yourself. See if you can find the best solution and run it by the owner. I don't know too much other than the horse will do better moving around versus being stuck in a stall.
But most of all:
have fun! Leasing was a great experience. In a way, it can beat ownership because the cost isn't nearly as high. it won't be as drastic if something happens and you cannot afford to lease anymore versus not be able to own. I pretended I owned the horse I leased...LOL!