The question wasn't "Is it efficient or cheap", it was how much energy is in a battery, and if and for how long would it run a fridge. If you also want to add one more point to why you probably shouldn't do it, car starter batteries don't generally like to be deeply discharged, you'd want to get a marine battery for that use.
As for how much the inverter would cost, depends on the fridge, but Amazon has a 1000W inverter for around $85, that should be enough for most. Ours could run from a 300W one, they cost around $30. Pretty handy devices if you want to run any kinds of electronics from a car anyway, I have one for when I want to charge my laptop and RC batteries on the field.