Whether or not cycling is allowed depends more on the type of road than how it is named.
Since you're already aware that "no cyclists are allowed on freeways" and you suspect that the entire N1 may not all be a freeway, you're half-way there.
Freeways in South Africa are designated by one of these 2 signs:
So if the road you're on does not have one of these at least every few km, then you're not on a freeway and the no-cyclists-on-freeways prohibition doesn't apply.
While the only stretches of the N1 I've spent much time on are close to Cape Town and are freeways, I can say that vast stretches of the N7 are certainly not freeways and I have certainly seen cyclists there.
So unless there's another prohibition preventing it, you should be allowed to cycle on the N1 on any section which is not designated a freeway.
If you'd like to start your trip closer to Cape Town, you could follow the R101 out of Belville as this road parallels the N1 for large sections and in some areas like the DuToits Kloof Pass actually is the original N1 route from before the Huguenot Tunnel was built.