You'll need to find the wires to the switch, why there are 3 wires instead of two, I unfortunately don't know.
I dont even remember #* though...
For what it is worth, here is a schematic of a typical rotary phone:http://www.freeinfosociety.com/electronics/schemview.php?id=1276
So that can give you a simple hint that the arduino can be made to dial a phone number simply by interfacing to the phone line with a simple relay switch contact.
We did this as a laboratory exercise on the old MMD1 microprocessor trainer. It used an 8080 processor and had a grand total of 256 bytes of RAM (that's bytes, without any prefix) for the program code.
Not sure that all land-line phone companies world wide still support pulse dialing, but they do here in the USA.