I run a small dorm fridge using an inverter. Solar charging keeps the batteries charged to provide the power.
The easiest way to calculate inverter size and how much power the fridge uses over a period of time will be to use a Kill-A-Watt meter. The KAW will give you a readout in total watts used over a period of time. It will also show the typical draw of the fridge when running.
Once you have the power measured, you'll need to get an inverter of sufficient size to not only operate the fridge but to overcome the Locked Rotor Draw of the compressor. The Tripp Lite inverters are known to endure a large initial draw and keep on chugging. Mr Wizard uses one with his 5'er. I use a Xantrex Prowatt 1000. My operating draw from the fridge is about 80W. Duty cycle is 1/3 -> 1/4. This all depends on ambient and how many door openings are involved. I initially tried a Samlex 600w inverter and it could not overcome the LRD.
I have 300 AH of battery capacity and 235W of solar panel.
Good luck with the inverter install.