A cut 'n paste from a not-finished faq I'm putting together:
Carryon Coolers?
Ok, you have a few rules you have to meet to bring it aboard.
First, NO ice! Ice can mess up the scanner and they will make you dump it all out before letting you in. Don't worry, ice is available on Deck 9 at the drink station and your room steward will bring as much as you want, up to three times a day!
Second, No hardside coolers(think 'Igloo'), those are what they are not allowing.
Thind, The size of your carry on must fit through the scanner to get into the terminal and on the ship after excursions. Disney uses an airport type scanner. The average size for one to fit is 22 inches long by 14 inches wide by 9 inches thick. It can't be much bigger than that.
And lastly, you can check beverages thru. They must be in a suitable container, I would recommend luggage or soft side cooler, but it has to be something that would contain any spills. Avoid using the grocery store bag

Again, no ice and also here more importantly, no glass.
(end of paste)
I would also insert a comment here - what are you going to bring aboard in said cooler? Any of it bottled water? That you don't need to bother. The water right out of the tap in your cabin is of equal quality to any botled water. All water from the taps/fountains on the ship is produced from seawater thru a process called reverse osmosis, the same process used in the 5 gallon home/buisness delivered water, and a lot of the expensive home water filter systems. You only need bring sport bottles that you can fill yourself with the previously mentioned ice from the room steward and cabin tap water. We do this when we cruise now.
Also of note there is an unlimited drink station on Deck 9 that has lemonade, iced tea, milk, coffee, makings for hot tea, and water and ice. You can also order down full pitchers of any of the beverages offered at the drink station and the fruit punch from the buffet to your cabin to store in the inroom cooler, for nothing more than tipping whoever delivers. And the cooler is powered; kind of like those ones you can plug in your car, it should keep whatever you put in it cool. So even bringing your own cooler is not that needed.
About the only thing you need bring yourself is soda and liquor. And even that, they have programs where you pay once and soda is free for the entire cruise, but you need drink a good deal for it to be a value, and a discounted beer program.