Until 5 may in JHB...no way!
Hire yourself a car, I would suggest a road trip, which can include a hunt.
My suggestions:
1) Hire a car and drive to Kruger National Park - it is out of school holiday, you will easily get accommodation. you can book on
https://www.sanparks.org/ as well. DO the south of the park (enter at Malelane 2/3 days) exit at Hoedspruit. From there drive to Hoedpsruit, Ohrigstad, but just before Ohrigstad (20km) turn left and do the Blyderiver canyon route (3rd biggest canyon in the world, biggest green canyon. Biggest being Grand and 2nd the Fish), from their Graskop, Sabie, Lydenburg Dullstroom and back to JHB.
2) Since you have time, I can also suggest still doing the KNP, but enter at most Northern Gate and travelling the entire KNP North to South, 5/6 days. Then you exit at Malelane and from their you drive Nellspruit, Sabie and do the Blyde. On your way North there are a million hunting farms/lodges.
3) Fly to Cape Town and do the wine routes.