I have not taken their New Waves course, but can give you some info about diving in general...
From what I've seen on other Princess cruises, the pool work starts almost at the departure time, and you should be finished with it in a day or so. They used one of the pools on the upper deck (the deepest one, as I remember).
The open-water dives would be like any other shore excursion - get off the ship, do two dives, then have the rest of the day available for other things.
You need 4 certification dives - typically get two per day. As for equipment, well, do you have your own mask and snorkel? If so, bring them. They recommend you bring your own snorkel because theirs have been in so many mouths. If you have a mask that fits great, go ahead and bring it too. Fins are easy enough to use anywhere so I wouldn't bother with them.
We did something similar a year ago - our 10-day Princess cruise didn't offer the New Wave (only on select 7-day cruises) so we did our classroom and pool work in Kansas City at a local dive shop, then arranged via email to do two cert dives in Cozumel and two in Grand Cayman. Worked pretty slick, and a good time was had by all. Now my son and I are certified.