You've asked a couple of questions and have raised a few issues.
Firstly, a solar power unit for a camping fridge is great IF YOU CAN AFFORD IT. The fridge itself (new) will cost a minimum of $1000. If you have enough disposable cash you could get a battery/ solar power rig but be prepared to spend upwards of $2000 PLUS your fridge.
You say you don't have a cigarrette lighter in your car ... is it an old car? Even cars that are 20yrs old have a cigarrette lighter. Note: you don't need the lighter itself, just the socket for the lighter as that is where the 12volt DC socket will go.
In regard to the length of time the camper fridge will last for on gas. This depends solely on the the size of the gas cylinder. You will need a dedicated gas cylinder for your fridge so if you need to get a new cylinder consider purchasing a 4kg cylinder or larger. Most gas ffridges will use up between 300 and 400mg of gas per day. That gives you approx. 12 days of continuous running and by that time you could easily be out of a non powered camping area and be at a caravan park where you can use the 240volt power supply and refill your cylinder.
The other issue you rasied was getting another battery for your 12volt supply. Again, are you REALLY sure there is no cigarette lighter? Even if you haven't a cigarrette lighter you could probably purchase one at Repco (or some other auto store) and have it fitted. A dual battery system is a great idea but complicated to set up as BOTH batteries must be charging when the car is running and therefore you need a competent auto-electrician to set it up which means ... MORE MONEY.
If you have a cig. lighter then you dont have too many problems and I would recommend a chescold fridge (50ltr) largely becuase of its size (only 50ltr gas/12v/240v in its range) and they are also a popular unit.
If you don't have a cig lighter I feel (my opinion) you should get one fitted as it would be the cheaper option and save a lot of problems in the long term. The K.I.S.S. formula applies here ...
I hope this helps.