Can I use an arduino as a ground short?
Depends on the voltage that is on that mute pin. If it's below 5 volt, you could use the arduino pin. You can measure the voltage using a multimeter.

If it's more than 5v you can use a NPN transistor and resistor combination. 

Current isn't a problem with these kinds of "settings" pins, as there is just a pull-up resistor that by default (unconnected) sets the pin to HIGH. By connecting the pin to ground you set the pin to LOW.