Cooling rate depends on temperature difference with an object's surroundings, among other things. All the specific heat capacity will tell you is how much heat per unit mass the material can absorb for a temperature change of 1 degree.
Well, the 1D heat conduction equation is
so a larger Cp would result in a smaller thermal diffusivity, and therefore an object with large Cp would respond slower to temperature changes (because of what SteamKing said).
If you assume everything is identical except the SHC and that "cools faster" means the temperature falls faster then the one with the lower SHC will cool faster.
If they both start at the same temperature the one with the higher SHC will start with more energy. If the heat loss in joules/second is the same for both the one with the highest SHC will stay hot for longer.