Fall or autumn.
Where are you planning on riding? The Cape is a good destination - wineries, whale watching, Table mountain, Robben island etc.
Joburg - rain should be stopping by then, but not really where I'd want to ride. Mostly flat.
Unfortunately my sister has now sold her B&B near Durban (East coast).
I can't offer too many suggestions - I didn't ride as much there as I do now. Just remember, Cape Town to Joburg is about the same distance as North to South California. (San Diego to north of San Francisco anyway) but with a lot less infrastructure/population. Most people think they can just go to Table mountain, then the Kruger National park, then the Drakensberg in a weeks vacation.