Can a MRI arthrogram on your shoulder hurt?

Yes, but... Not that much. When done by an experienced Doctor, the arthrogram should not be very painful. I have done over 15,000 arthrograms in the last 10 years, and most of my patients say it is not much more than a typical flu shot. You can google "DrNefcy.com" and go to arthrogram advice for more information.
Possibly. The injection prior to the MRI can be painful. The MRI is not painful, but sometimes with painful conditions it can be difficult to lie still. You can also have claustrophobia while having your MRI scan.