I'll be in the same "boat" on my next cruise. Since I'll be bringing our baby on the cruise. I'm not planning on doing much pumping -- I'll be giving him milk "straight from the tap." Since our son will be 8 months, I'll bring a little juice for the night we hope to have in Flounders. However, if I do pump any milk, I will be packing it in ice and then putting it is the room cooler. I'd have the steward bring in ice every morning and evening and put the ice into the cooler. Maybe bring a smaller softsided cooler to put the ice into, then pack the breast milk (I'd store it in breast milk safe bags) then pack these bags into the ice. If you change the ice before it completely melts, the breast milk should be cool enough.
If she is exclusively pumping and either sending back the milk after the trip or needing to store lots of milk during the trip, I'd ask the steward if you can store the milk in the ship's refrigerator. When I travel without baby, I store the milk in freezer safe bags (I use Bailey bags), label with date and volume, and store smaller milk bags in larger freezer safe ziplocks (labeled with name, date, and room #). To make it easier, I'd have the steward bring it to the freezer every morning & evening or just every evening.
Let me know if you need me to post the storage guidelines at refrigerator/freezer temps or if you have any more questions.
Good luck,
Jennifer