Hi Antti,
To be honest if you are moving I would really try and buy a car. The rental cars that are the older ones are usually rented by students and are NOT well kept. If you are in an accident you will lose your high deposit, and be without a car.
When I first came here I looked around for rental cars and if you can drive manual they run R2,500 a month. I think it's best to try and save for a car. I bought a car wasn't very good but it lasted for 2 and half years for R18,000. Yes it was not the best. But, it was an automatic little car that someone needed to sell asap as they were leaving the country.
I guess the point is see if you are really saving that much money by renting one.
As for leasing I can not be much help with that sorry. I just like to save up and buy a car.
Check out these websites
Gumtree.co.za
and Autotrader.co.za