If it's the same as on the other ships, no. I asked maintenance about the temperature once and they told me that they keep most things in the stateroom turned off during turnover on the day of embarkation, so the beverage cooler is at its warmest when you first arrive. If you fill it with beverages, it will cool off a bit overnight that first night. It won't get American refrigerator cold but it will chill canned and bottled drinks alright. I always sail with drinks in the cooler, and from the second day onward, they're always cool enough for me, and I don't like warm drinks.
But milk or milk products? No, I would never put them in there, at any time of the cruise. I tried to put a dessert from Cabanas in for several hours once (not ice cream, just some other chilled Cabanas dessert), and it got ruined in there. In a regular refrigerator at home it would have been fine.