L.P.
We did a cruise last fall, our youngest was not quite 3 yrs old at the time. We had a blast. I think as long as your baby will nap in the stroller you will be fine on excursions. We took the stroller with us and used it on most excursions, our son was still napping and would either nap in the stroller or on the bus rides.
At dinner just ask and they will have a highchair at your table each night. The cruise we were on even had a kids menu so you may be able to able to order food for your baby (such as plain pasta).
We even took our boys to the shows at night on the ship, several nights the youngest was sound asleep before it even started but he just slept on our laps.
Someone else mentioned the cab rides with no carseat. I'm not going to say I would advise it all the time however, living in a big city where I can't carry a carseat with me at all times, I have done it. We go on many vacations where we don't get a rental car so we don't take the carseats, we have done many cab rides in cities and just had the boys in the car with us. When the youngest was an infant I help him on my lap, now he seats in the seat.
You will have a great time!