Power = Voltage x Amperes
= 115 x 1.4
= 160 Watts approx
Let us assume 200 watts to be on the safe side.
307 KWH per year is about 0.8 units per day and assuming that the compressor runs for 1/3rd time though the fridge is on for 24 hours, we get the power consumption as 100 watts per hour.
In case of AC appliances there is a power factor and it may be about 0.7 or so in this case and thus the watt rating and the power consumption will differ. We need to take the watt rating into consideration for calculating the instantaneous power needs.
Inverters of 400 - 600 watts are easily available (look for an used one in working condition if possible).
How many hours it lasts depends on the size and condition of the 12V battery. Look for the AH number. Divide that by 12 to get a very approximate value.
Using Ohms Law: Power =Voltage x Amperage. 161 watts = 115 v X 1.4 amps running. Startup power for the compressor will be higher ( as much as 10x).So, to be safe; get a 1000-watt DC-AC convertor.
As far as how long the battery will last. The time period it will last is dependent on amp-hour rating of the deep cycle battery. The battery's label will list this number. Use this web calculator to estimate the time it will take for the battery to be 50% discharged. http://www.solarwindtec.com/batcac.shtml
You want to make sure to use a deep cycle battery, as a regular starting battery will not hold up to the contant discharging/charging cycles.
You will need to know the amp-hour rating of the battery to judge the running time The inverter is going to pull more than 14 amps - probably 18 or more depending on the efficiency of the inverter. And once the battery voltage gets low enough, the inverter will shut down completely.
You will, for anything more than a half days use, probably have to get a splitter (from an RV place) so that the battery is charged from the vehicle circuit but does not run down the vehicle battery when low.
You would be better off to buy a 12 volt RV refrigerator.
The KWH/year is irrelevant.
Your fridge pulls 161 watts because 1.4A * 115 V = 115 watts (P = V*I)
You'll need at least a 200w inverter, if not bigger.
Golf cart battery. Not big enough capacity wise. You will need a large deep cycle battery like is used in a boat to keep this thing going. A deep cycle battery lets you pull it way down before you recharge it, but when you do recharge it, you have to recharge it slowly--totally different from a car battery. An older large car battery would do the trick as well.
I'd guess you'll need about a 300 watt inverter. The refrigerator will run the battery down in about 3 hours or so. These are rough estimates.