Here are a couple of things to consider.
There are only 2 nonstop direct flights to Johannesburg from the U.S. Delta (out of Atlanta) and United (out of Newark). Its a 15hr-ish flight and even with upgraded seating I found the flights to be brutal. Easily the worst part of the trip. That being said, I still much prefer that over connecting flights out of europe or northern africa. This will add hours to your trip.
You also need to consider how close to Johannesburg your outfitter it. Do you have to take a connecting flight to Cape Town? Or are taking ground transport.
The United flight (assuming no delays) will arrive around 5 in the evening. That is still early enough to catch a connecting flight to Cape Town or ground transport on to the outfitter.
The Atlanta flight arrives around 7:30 in the evening. Unless you are fairly close to where you will be hunting, you likely are spending the night in a hotel in Johannesburg and driving up in the morning, which means most of your first day hunting is shot. You will be driving up in the morning. Once you arrive, there's settling in, lunch, and sighting in your weapons. Also don't forget you are in the southern hemisphere, which means its getting dark at 5;00-5:30 in June/July.
If you have a connecting flight to Cape Town, the last flight is usually around 9-ish. Thats cutting it close. Don't forget you will have to recheck your bags to take a domestic flight. Better hope there isn't a delay.