Jen - we did this cruise last December - fantastic.
Dress is layers, with a wind and waterproof outer layer. That should do the trick. Don't forget gloves,hat & scarf. We didn't find the weather too bad. Warm in Cape Town then cool getting a bit colder, but nice. There are always locations inside the ship you can have a great view without going outside in the cold. Evenings, just make sure you have a jacket or sweater as usual dinning rooms can be a bit cool. Don't forget your bathing suite and make sure to have slippers & robe (request a robe for your room) to put on when you get out of the hot tub. There is also a hot tub inside. We didn't use the swimming pool on this cruise.
We also love the "Royal Princess" and are booked on the 21 day cruise, April 21, 2005 cruise, Manaus to Rome. That's the next to the last cruise before she goes over to P&O in Great Britain. We had also checked out some postings on Cruisemates.com as well as Cruisecritic.com for some advice before we sailed. I know you'll have a great time. Bring lots of film or a digital camera with extra memory. We bring our laptop computer along to download pictures each day. Feel free to contact me if you have any further questions. Happy cruising, Lorraine