If I am understanding this correctly :
The US$2.99 DX board is a manually controlled MP3 player.
You want to integrate that into a robot and have it controlled by an Arduino.
It MIGHT be possible. I think you might need to buy 2 or 3 of the board's for destructive testing.
I would think the first step would be to desolder one of the buttons and see if you can get to the traces on the board.
If you can, then soldering in some wires and having an Arduino write HIGH to simulate a button press might do it.
Require some messing around and tinkering.
Cheap solution if it worked, though. Intriguing.