Shower head arm rusted through
You probably can fix this without having to damage the wall. This is a common problem, and there are inexpensive tools designed to remove any pieces of the shower arm left inside the wall in the fitting. 

You might be able to borrow one, but similar tools can be bought for less than US$10. Check your local hardware store or plumbing supply store.


Unscrew the shower head. A piece of rubberized shelf liner or jar gripper is useful to protect your hand and provide a high-friction grip on the fitting, but pliers can be used, at the risk of damaging the shower head.
Using the rubber gripper, again, turn the shower arm counter-clockwise to remove it. It may break off, or pieces of the thread may remain in the female fitting, and this is where the internal pipe extractor is useful. Be gentle, since you do not want to damage the fitting in the wall.
Replacement shower arms are no more expensive than the extractor, and are common hardware-store items. Measure the depth of the hole, and bring in fragments of the shower arm to get the correct size part. A good brass fitting should last many years.


BTW, I've removed pieces of a brass fitting with just a flat screwdriver, inserted at an angle -- but the old fitting was in much better shape than that in the photo.