As far as I’m concerned, there’s only one place to stay in Jodhpur, especially if you’re a foreigner living in the city on a long-term basis. It’s called Indrashan Homestay, and it’s basically a bed and breakfast, run by an impossibly friendly couple named Bhavna and Chandra. They mostly accommodate short-term guests, but they also have a special place in their heart for long-term visitors. The rooms are comfortable, the neighborhood is quiet, and the food is the best in Jodhpur. You kind find more information on their website. Plus they’re very highly rated on TripAdvisor (try to guess which review is mine).
If you absolutely have to stay in the old city, you’ll find plenty of decent guesthouses at very reasonable prices. Go to Lonely Planet for a comprehensive and reliable guide.
Finally, if you have more money than whichever deity you choose to worship, then you don’t need me to tell you where to stay. There are several outrageously opulent options in Jodhpur that are not very hard to find.