I'm not the best person to answer, since I have only been to Cape Town a couple of times, but since no one has answered, here goes what I think:
You certainly can't walk all the way, specially if it's in the evening. I have always rented a car when in Cape Town and, as far as I know, it is the best way to get around town. Taxis are not that easy to hail on the street. You are better off asking in a restaurant or hotel for help in ringing one. You can also get a few cab services numbers at the information desk in the airport.
Good luck! about 5 years ago
It's about 3 km from the port to the CBD, so a taxi is a good idea. It depends where your ship ties up, but if it's in the V&A Waterfront development it'll be easy to get a cab.
Make sure you get a metered taxi - they're run by the city and are safe and reliable. Don't use "gypsy" cabs or unofficial cabs.
Most hotels in Cape Town offer a pick up service at the airport, perhaps they'll do the same for you at the port? about 5 years ago