It depends on how the buttons are wired on the original. If you want to make sure there's no interference with the original guitar-shaped controller, you probably will need sort of solid state relay or opto-coupler. If you know have more ideas on what kind of voltages (I guess either 3.3V or 5V) and the polarity used on the buttons you can probably also use simple mosfet. But don't forget in that case the common ground between the Arduino and the the controller.
Well, I found that very interesting, but had no MOSFET...but I made it work with a simple transistor BC547...I´m probably frying something, but sofar it has worked just fine! This is the schematics: