The ATF isnt going to have a lot of information on this.. what you need to be looking at are the import and export laws/regulations for both the US and SA.
"exporting" a hunting rifle has gotten a whole lot easier over the past 12 months.. the export of all firearms used to be controlled by the International Trafficking of Arms Regulations (ITAR), which is controlled by DDTC at the US State Department.
Recently that has changed though, and the export of firearms that are not military by nature are now controlled by BIS at the US Dept of Commerce (via a lower set of regulations).. My experience with BIS vs DDTC is that they are generally slower to process paperwork and a little more difficult to deal with.. but that less overall work is required to apply for an export license through BIS..
Youre going to have to apply for a permanent export license.. you can do that on your own.. but it would probably be easier/better to hire a broker or FFL that has experience doing this.. If your paperwork is not exact/precise, it will almost certainly get rejected and have to be re-submitted.. and the process can take a substantial amount of time from beginning to end..
And the recipient of the firearm is going to have to get permission to import the firearm through whichever governing body in SA handles such things..
Here's a starting point for you:
https://www.bis.doc.gov/
FWIW, I am NOT an import/export expert.. there is probably someone here that has a far better understanding of what is required than I do.. I just work in a field and for a firm that exports a lot of regulated/controlled stuff... and oversee the guys that do this for us..