there's a "fridge cooling" thread in this forum where people say that a mini fridge isn't designed to run continously and keep your case cool, and it will quickly lead to failure of the fridge.
Personally, I think it'd work, my mini fridge was able to run as a freezer before i gave it away. I'd suggest leaving the power supply outside, and have some kind of dessicant inside the case to avoid condensation.
and good luck.