Is my Outdoor Mini Fridge Doomed? Wont get cold
The only moving part in a mini fridge is the compressor.  If you plug the uint in and hear something come on, that is the compressor.  If the compressor is running, the wiring is probably ok.  

If the compressor runs for a while and the fridge never cools, then you most likely need a professional to check out the sealed refrigeration components.  Normally this isn't worth the cost on a mini-fridge but outdoor units are much more expensive so it's probably worth a repair call to check it out.