Spell check?
No, its not going to get anywhere near cold enough to make a difference. Think about what you are planning:
1. Fridge cools radiator
2. tubing carries coolant to 2nd radiator on case
3. 2nd radiator has room-temp air blowing through it, thus warming the 'cool' air.
4. process repeats.
To be honest, similar ideas have been posed several times before. You would be far better off getting a cheap window air conditioner and just blowing the cold air into your case than trying a loop like that. A minifridge would burn up trying to continuously cool the coolant, even if it remains close to outside room temperature. Refridgerators are meant to remove heat ONCE, then maintain cool temps...not consistently cool.