Every cruise line I've been on (I've only been in balcony staterooms) have had fridges. I get to the room and take out all the crap they have in there (to try to entice you to spend money) and put my stuff in there. Mine kept things cold.
If you need something to keep medicine cold, ask the attendant, they may have something you can use to keep it cool. I know if you write ahead of your cruise, they will make accomodations for you. Good luck!