There exist a switch that goes between the power usb of the RPi and the power supply. When you turn it on, the RPi boots up. Then you can put a script into the right spot (depends on the flavour of the OS) and your MP3 player will start running.
When you shut down the RPI, it still consumes power and you'll have to power cycle it. (and if your power supply remains plugged it, it is consuming power, too)
I, too, would be interested in a technique to start the RPi after a shutdown, other than power cycling it.