On shorthaul, bottles of champagne, whether large or small should be loaded in the top drawer of the bar trollies. There is a slot/shelf at the top of most bar trollies where small bags of dry ice are loaded. This should in theory help to keep the bottles chilled. It doesn't always; and catering sometimes forget to load dry ice.
Shorthaul aircraft are not fitted with chillers in the front galleys. On the shorter flights, where the small bottles are loaded, it's possible to place a few bottles onto ice in the insulated boxes containing ice cubes - but this isn't an official method.