Learn about your Arizona Tenant Rights.
If you have a valid written lease and can prove you paid the owner of record on the date he owned it you have a good claim to occupancy. The five day notice to vacate must be followed by a court order. If your leases expires in only 4 days I don't' see how the legal system could move that quickly. As far as the A/C repair goes I don't think you should expect any solution before you leave. This is one you may just have to chalk up to experience and add it to your war stories.
Best to try and work it out with the new owner-he doesn't want to pay the legal bill anymore than you do. If not you could call a real estate attorney and get some advice.