As a visitor to Cape Town, if you wanted to cast a line by yourself (without chartering a boat) then I guess it could be difficult to know where you may fish and where to purchase a fishing permit.
Our government's Marine and Coastal Management website has all the information that you will need to know: where you may fish, what you may catch, how much you may take home and what the permit will cost.
In general, licences currently cost between R40 and R70. In the past I've bought a licence from a local Post Office, but if you have any trouble figuring out what to do, don't hesitate to give the client call centre a call on +27 86 112 3626 - they'll be able to give you all the info you need.