We were on the Fantasy a couple of weeks ago and my brother- and sister-in-law had their 15-month old with us. They brought a decent-sized Peg Perego stroller (not huge but not an umbrella either) with them and used it quite a bit on the ship, including having it in the MDRs and Cabanas. It never really seemed in the way in the dining rooms or out around the ship, with the possible exception of waiting for elevators. People use strollers on elevators in buildings, so I don't see any problem in having them on elevators on the ships, as long as you understand that it may require a slightly longer wait to get on an elevator. The rest of the kids in our group (10, 10, 8 and 5) usually preferred taking the stairs, so a couple of us adults would go with them, while one or two others would stick with the stroller.
With that in mind, for a 4-year old, if you feel like you need a stroller, I'd definitely stick with the umbrella version -- much easier to maneuver, definitely easier to find a place in a tiny stateroom for it, and you don't really expect a 4-year old to use it as a de facto bed as a baby might. Even with a stroller (whatever version you use), I think you would find that you really wouldn't use it that much on the ship. It's a big ship as far as cruise ships go, but it's still a far cry from WDW walking! The average person might walk five miles at WDW on a given day, but rarely are you required to walk far on the ship. The only real comparison to being at WDW might be standing in line for a character greeting, but even then, there are often other seats nearby that a kid could use to sit in other than a stroller.
One place that you might find a stroller the most helpful would be on excursions off the ship, as that might give you a more secure feeling with the child -- i.e., less risk of running off, but also dealing with a long walk off the dock into the port, especially on the way back to the ship if he/she is worn out.