Tell him what your intentions are. If you are looking for someone long term to shadow and he is unable to, or unwilling, maybe he can recommend another physician that is more open to the idea. When shadowing a DO for an eventual LOR, it makes the experience so much better if you're paired with a physician who enjoys teaching, as opposed to one who you get the feeling you're wasting their time.
I was lucky that I found a DO who was younger, and more enthusiastic about letting me shadow him once a week in his practice and the hospital, as well as spending lots of time telling me what to expect in medical school. It helped that he just started his practice so he wasn't as busy as a more established doctor.
As far as questions, ask him about omt, if he uses it, why or why not, his opinion of omt, if he regrets going into medicine (I'm amazed how many doctors tried to tell me not to go into medicine), if he drives a mercedez or bmw etc.