If the 12 volt car compressor is the type for inflating tyres, then the simple answer is, no, it won't work.
As Andy said, you don't need to spend a fortune to get a reasonable airbrush and compressor. I've been using an AB dual action, fine line brush for the last four years or so, with an inexpensive, simple compressor, and the results are every bit as good as the most expensive, all singing, all dancing items. OK, the compressor doesn't have the versatility of some, being 'fixed' pressure, no gauge etc, but it works, and it didn't cost the earth. (around £40 if I recall correctly, with the 'brush at £28).
There are normally some package deals around, where you'll get a reasonable 'brush, hose, compressor etc for around the £80 mark, or items can be purchased separately. have a look at Model Hobbies web site, and also the following (look for their own make - 'AB' - 'brushes.) :-
everthingairbrush.com
These should give you some idea of what's available.
Hope this helps.