When you know how to write software, then it is possible.If you are new to programming, then Arduino is a good way to learn it, but not while trying to make this project.A servo motor is slow, but easy to use. I would use 4 servo motors, one for each location.It is a capacitive or resistive touchscreen ?What kind of device are you operating ? Perhaps that device has a serial port and accepts commands.
1000's of times in how long? Is that a calibration routine?
what sort of touch-screen technology is used?A capacitive touch screen needs more than just touch....R
Probably over the course of a week or two? Not a good idea? Processor not strong enough?
It's a capacitive touch screen. Any ideas for a finger alternative?
Just out of curiosity I tried an experiment with my old Samsung phone. Wearing a rubber kitchen glove the screen would not respond but a piece of hex aluminium bar held in the glove would cause a response - for example selecting an image in the gallery. The piece of aluminium I used is about the size of a 15mm cube and I was touching the flat face to the screen.Note that a metal probe might scratch the screen....R