To use any device with buttons, you could use reed relays parallel to the buttons of that device.
The Arduino is good with sequences and controlling things.
Add one or more push-buttons to the Arduino, and make a sketch that runs a sequence with the reed relays if a that push-button is pressed.
Start with an Arduino Uno, that is the easiest Arduino board.
But reading about your post.. What if you can place all the tunes on a micro SD card (up to 16 or 32GB). Use an SD shield for the Arduino, and let the Arduino play the tunes.https://www.sparkfun.com/products/9802
You could easily make 2 or 3 of of those. So if something is broke, you can swap it for a new one.
A Real Time Clock could be added to the Arduino. Or perhaps a LongWave time signal, but that depends on the country you are in.