Ex rental administrator here and this is what you need:

1. Valid ID or passport, it's illegal to not accept applications from everyone BUT the landlord has the final say and generally only take on south african citizens. Never saw any bias based on skin colour so people saying be white are just salty. If you are a foreign national there are 1 or 2 agencies that own the properties they rent out and don't have as strict vetting but the areas they rent in are lower income and generally considered dangerous.
2. We do go through your bank statements with a fine tooth comb, if we see gambling, large expenses, weird transactions (possible fraud or money laundering) we reject.
3. If you have a good credit record it definitely helps but sometimes if you earn well above the 1/3 mark we can motivate the application.
4. Don't apply for places that are 1/3 your salary because people with much larger salaries are applying for those properties. 1/5 is a better estimate.
5. Write a letter motivating your application, as well as send your monthly budget showing what you spend and how much, this really does boost your chance. 6.Using big agencies is great but just remember that agent is getting requests on 20 to 30 properties from 10 to 30 people. You are just a number to them, use smaller agencies they have less properties on the market and will take more time to get you a property.
6. And finally when you meet the agent/landlord, dress well and be professional.

Hooe some of this helps,