Maybe someone else can give you some suggestions... I can't help you with that. The only way I'd attempt is if it was working well-enough that I could replace the buttons with Arduino-controlled relays.
Did the boombox play MP3s before? If not, you'd need an MP3 decoder chip (or something more powerful than the Arduino). And, you might need a different controller chip for the drive (accessing a computer file is different from accessing an audio track).
My internal DVD drive of my laptop isn't working at all
You don't need a CD/DVD drive to play MP3s. If you have access to a working computer you can rip the CDs to MP3 (or FLAC, etc.).
Or you can replace the broken drive for less than the price of an Arduino, and you'd be done the same day you get it!