Driving is easy (though easier if someone is map reading for you). We did a tour of the Western Cape over 2 weeks and didn't get as far as Addo, but I believe it is well worth doing, so it worth missing out on other bits - you will want to go back!
I'd obviously still have time for other things if I did this
I think even if you stick to the main trunk roads between places of interest, you won't have any spare time with this itinerary. It's good, there are good choices of places to visit, hike etc, but if you want to drive both there and back then after taking into account travel time and you also want to experience the place, then you have enough pencilled in. Assuming / recommending you spend at least 3 days in Cape Town at the beginning and at least 2 nights at Addo.
From your itinerary I assume you like the great outdoors, and the whole area has fantastic scenery and outdoor activities throughout. You can easily spend 7 days in and around the immediate vicinity of Cape Town hiking, mountain biking, white water rafting, sea kayaking and so on. I really like wildlife watching and scenery. For us the highlights were:
Cape Town itself, taking in the history and city, hiking up Table Mountain and/or Lion's Head (make sure you have a few days in CT and get up TM as soon as you have a clear morning), driving round Cape Point with sunset at Oudekraal or sundowners at Camp's Bay.
Betty's Bay penguin colony, whale watching by boat in Hermanus (you get very close) and whale watching from white sand dunes at De Hoop. De Hoop is a bit of a detour, some of which is over gravel roads which takes some time, but the whales were fantastic, displaying and communicating over the whole bay. Of course we could have just been very lucky!
I loved driving the R62 through the different climatic zones between the grenery of the coast and the arid Little Karoo. The Outinequa Pass (Oudtshoorn to George) is very pretty and you get a look-out when travelling south. Though as said with it being winter it is possible it will be clouded in. There are natural hot springs at Warmwaterberg on the R62 with very basic accommodation.
For information on great hikes and accommodation in reserves, see Cape Nature.
As well as August/September being the best time to see Southern Right Whales, it is also the flowering season for the Proteas and wildflower (daisy-type) meadows. The Kirstenbosch botanical gardens should have lots of colour and nectar-feeding birds around.
Enjoy, it's a fantastic part of the world.