The all aboard time in Port Canaveral is 3:30 or 4:00pm so 5 is not that much later. I think that the reason for the later departure is that St. Kitts is not that far away - will be an easy cruise.
A few years ago we did the Southern out of San Juan and our first stop was Antigua. All aboard was 9pm. Then a sea day, then to Antigua. We were barely moving - not much over 10knts, probably slower. St. Kitts and Antigua are pretty close together. So we will probably be traveling closer to normal speed to make St. Kitts the first morning.
Back to San Juan. We spent a day in San Juan in January and it was 85 degrees. Not too hot, but a lot warmer than at home. Carry water bottles! Bring Sunscreen.
Don't know the ages of your kids, but we love Old San Juan. There are the two Fortresses. They are both part of the National Parks Service. I think that the price of Admission is $5 per person, good for 5 days, and covers both locations. You can spend some time in those.
There is a paved walkway at the foot of El Moro (one of the fortresses) at sea level (well 3 or 4 feet above sea level). Fun walk, get to see the feral cats and the iguanas. It's a nice walk with the sea breeze.
A few years ago we did a food walking tour of OSJ which was a great overview of the area and tasty too. There are other guided tours of OSJ. I would suggest booking one of them for your first day there to get that overview of the area, and some history.
We went kayaking at the bio bay in Fajarado one night - really cool - the water lights up as you paddle through it. Booked this through our hotel along with transportation.
We spent a day at the rain forest. Again booked the tour through our hotel. There are many different tour groups you can book through. Was very interesting. I think that I would stick with a tour rather than renting a car and driving. Not much if any signage, thought we might be doing some off roading, etc. I think our driver knew a "shortcut" in and out of the rainforst.
If you go the car rental route I would suggest getting the rental car company's Physical Damage Insurance / Waiver. There's a high probability that it would come in handy (but I have been to places where they drive way crazier).
I hope that we are able to come into San Juan a few days early again - I really like OSJ, and I didn't know that Rice and Red Beans could taste that good. Will probably get out of the city and go further afield next year.
and yes, I'm booked on the same cruise as you are.