Any day will work and it doesn't matter if you shoot the Dog
While the achievement is pretty self-explanatory, there isn't a strategy to acquire a golden shell. However, I did find it easier to do this on medium/hard.
Before you start the round (on any day) make sure you have at least one normal shell in the chamber,then just keep firing shots until you load in a golden shell. Once the golden shell is loaded use your normal shell to start the round and shoot the sun with a golden shell.