My advice when dealing with kids on these port intensive days is to keep them hydrated, let them sleep during transporation, bring breakfast to the cabin for them if they want so they don't have to get up quite so early. Lay their clothes out for them so it's easier to get up and dressed. Sunscreen, hat (if they'll wear one). Some portable snacks even if it's a box of cereal off the breakfast buffet - we take a lot of froot loops in our bag.
While you want them to take everything in, remember that they aren't going to be as interested as you so if they are drifting off or staring into space, don't worry about it.
We talk to our kids ahead of time about behavior and how, even when they are tired, they still need to be courteous and polite to us, to each other, to the other guests on the tour, and to the tour guide(s). And that we will be the same towards them.
And when you get back on ship, don't push dinner if they don't want to go - they may want to run off and play with their new friends or youth programs - after all, for most of our kids, it's more about being on a cruise ship than it is about what they'll see or do in the ports.
Try to find something fun about each excursion - maybe get gelato at each port, or have a particular souvenir to collect - that's how my boys started collecting thimbles and spoons - we started in Capri in 2007. It became a quest. Still is.
And on the sea days, let them sleep til they wake up and you absolutely HAVE to get them up so your cabin can be cleaned. We have had to enforce a curfew on nights before early excursion times because we knew they needed to be more rested - they quickly learned that the extra time sleeping was much better than the time spent doing activities.
So what about you? What advice do you have for dealing with children on port-intense days? Post your tips in the comments.