If I am currently living in South Korea as a South African and intend to travel to Hong kong, do I need (could I) get a visa at the airport in Hong Kong if I want to go to Beijing as well?

It's true that South African citizens do not need a visa for Hong Kong (for under 30 days), but you need a visa to Beijing. You can try to call one of the designated travel agencies to enquire if the ARC with at least remaining validity is still required. If you really can't get a China tourist visa in South Korea, you can go to Hong Kong to have a try, in fact, it is not easy for you to get a China visa in Hong Kong because you don't work or reside in Hong Kong. There is an exception, it is possible to get a SINGLE entry China visa valid for 15 Days in HK, provided that all your documents are upto date: that u have previously travelled to china on a valid visa before; ur passport has no cancelled china visas and that you have a confirmed return ticket or a ticket showing exit from China and a confirmed hotel booking voucher.