If you have not already done so, I do recommend looking at the web site for AZ Game & Fish. If you are doing this, you might as well also get a mountain lion tag.
Are you wanting to chase them in the spring or the fall? I ask since they are two different hunts.
In the spring you never know when the bears will come out of hibernation. This can make it a challenge when scheduling a hunt; that and combined with the female harvest limit.
In the fall, you need to be prepared to high heat and potentially high humidity. Also, you will need to be prepared to quickly break the bear down, get the meal cool, and get the fat off the hide. While you may want to hunt them in the mountains, they can be down much lower eating prickly pear fruit.