There are several factors that come into play. First you do have to get your SAPS permit but it will be an "In Transit" permit. Same form just one box checked differently. The next factor is whether or not the airlines will play nice and check them through on your return trip.
On my trip to Botswana this year I got my permit and took possession of my rifle upon arrival in Joburg, spent the night there, and checked my rifle with the second airline the following day. On my return trip I was able to work with the first airline to ensure that they would check the gun through and deliver it to the second airline. At each step in the process the airlines were able to confirm "yes we have your gun and this is what we are going to do with it". Pretty painless really.
I have always been a DIY guy when it comes to travel. I have never used an agent for any aspect of any trip. However this year I used riflepermits.com and it was money well spent. Being able to walk into the SAPS office pick up your gun and your permit and leave within 5 minutes is nearly priceless after the long flight to Joburg. I have spent up to 3 hours in that place in the past.
Good luck, relax, and enjoy!