It certainly seems like a possibly good arrangement - I am guessing from your use of 'ranch' and 'trails' etc that you are not in the UK? So it is hard to say whether it's good value for money compared to what a similar scheme would cost in the UK - here we would probably call it a part-loan or share rather than a 'sponsorship' which would be more of a donation to the charity rather than a reciprocal arrangement. But as you say it is a great advantage to just be able to ride without worrying about stable duties or the other variable expenses of keeping a horse e.g. vets bills!
Couple of things to think about (don't feel you have to answer the questions, just what my thought process would be in weighing up the decision!):
-You say you could ride around 3 times a week if you went for this, do you actually have the time and inclination to commit to that much riding, particularly in the winter/bad weather (if you get bad weather where you are!), or realistically would it vary a bit? If so does the centre have an issue with that, i.e. are they counting on you exercising the horse 3 times a week or is it simply you pay your money and can choose to ride as and when? If you end up only riding roughly once a week it would be much cheaper to just pay for the rides you have rather than this sponsorship arrangement (although if that has an element of contributing to riding for the disabled that is obviously worthwhile in itself).
- How restrictive are the times you can ride the horse? In the UK when riding schools offer this kind of share they usually stipulate you can only ride at 'off peak' times when other clients aren't using the horse which can be inconvenient. With your arrangement, do you have to commit to riding at the same times each week or can you (within reason) turn up whenever? How does this fit with your lifestyle/other commitments?
-Can you choose which horse you 'sponsor'/get to ride and are you guaranteed the same horse every time or can the centre substitute at will? Again many riding schools in the UK which do this often only offer their less 'useful' horses for the sharing scheme, i.e. ones which are larger or smaller than the majority they need for lessons, or are more green or difficult to ride, so they don't get as much exercise/use from regular paying clients. I'd be particularly careful about this as you are a somewhat inexperienced rider so would want a steady, safe horse, plus it's nice to build a relationship with 'your' horse rather than riding lots of different ones. On the same note will there be any help/supervision available when you ride or do you literally saddle up and head out? Are there others around for you to ride with? Is there an arena/school you can take lessons in if need be or is it only trail riding? From a safety point of view it would be better not to go out on trails completely alone at least the first few times, plus horses which are used to being taken out in a group ride/trekking type setting (which I assume this is?) sometimes don't transition well to being ridden alone so you would need to bear that in mind...
-What happens if 'your' horse is injured and can't be ridden, do you still have to pay? How much notice do you have to give to end the arrangement?
-If you were in the UK, you would need to factor in the cost of personal rider insurance that covers you for riding in a private arrangement (as opposed to in a school) to cover your for personal accident and 3rd party liability. It's not very expensive in the UK (comes free with BHS gold membership). Not sure if similar things apply where you are but worth looking into?