How do I build or buy a Servo Motor that does this?

"Hi, how do you know what it is connected too, all I see is the servo working tapping the screen.
It looks like a switch in the end of the arm detects when it has touched the screen."

...Where I originally found the item, it shows it powered by simply a USB end.... it can be plugged into your computer for 5 volt power.

"The end does not have to be conductive, these touch screens use pressure between transparent screen layers and their resistance to indicate an operation."

...Yes, the iPhone screen must have a conductive "thing" touching it in order to control the touch sensitive screen (i.e. why your finger works, and styluses work, but not a piece of plastic or rubber.) The early touch screens used pressure sensitive response between screen layers, but they were much less sensitive than the new technology.