Yep it's all possible. You will have to be able to master a series of task before it's a done deal:
Find a landline phone and learn how to supply it's off-hook DC line voltage so that it will have power to be able to send it's touch tones as you press it's buttons (called DTMF signalling)
Get a DTMF decoder IC or module (or possibly a DTMF shield, someone must have designed one by now) and have it wired to the phones analog 'local loop' to extract the specific audio DTMF tones into specific 4 parallel bit binary values as the phones buttons are pressed.
Read these values in the arduino sketch
Detect the desired number or number sequence to activate a give tune
Use a wav file shield to output tunes under arduino sketch control.
Your difficulty will be directly related to how much or little electronics knowledge and experience you have, but it's not a particularly difficult hardware project. The software would be pretty simple.
Lefty