How do I make a micro-controller act as a finger on a touch screen?

Yes, this is quite possible to do. Simply place a small piece of copper foil on the screen. Connecting the foil to ground will simulate a finger press. You can use an N-channel FET to switch the foil between connected and not connected. The gate of the FET goes to your microcontroller.

This can be done if you get an Arduino plus a servo controller that has a rig that controls a stylus. The stylus must be able to work with capacitive touch screen (not all do).

Look for Arduino Servo Controller shield