Your hunch is right - that adaptor will not work in South Africa (it will, however, work for South Africans travelling to other countries: it has the right round holes to take a South African plug but not the prongs to plug into a South African socket).
I have the exact same type, and some similar models, and needed to buy special South Africa adaptors. It's the first country I'd been to where they wouldn't work. You need one with a plug that looks like this - three fat circular prongs in a similar (but not compatible) arrangement to a UK plug.
It's bulky and I've never seen a multi-country adaptor that includes this type - they might exist, but it's probably easiest to buy a cheap South Africa adaptor then plug your USB adaptors etc into it.
If not, do the major cities have outlets for other countries (US, UK, Europe)?
When I was in South Africa, the big international-customer-oriented hotel I stayed in Johannesburg had international plug sockets, but of the other places I stayed, if I remember correctly it was only one in ten that did.
I'd recommend buying a cheap adaptor online. They're not hard to find when you're there, especially in airports, but they'll be cheaper when you're not a captive market.
They're mainly used in South Africa (plus Lesotho and Swaziland and some old electrics in formerly-British-influenced places) so just search for "South Africa adaptor" and you'll find something.