The rum gives Captain Morgan some sort of power when he drinks it. However, in the final battle against the ninjas, Wario makes the observation that cereal-eating pirates fight better than the rum-drinking pirates.
As shown by Morgan in "The Super Mario Brothers," it gives him the "drunken fool" ability. This is similar to the Super Star from Super Mario Brothers. This allows Morgan to charge through the ninjas without getting harmed. Later on, he tries the ability Super Burning Lava Hot Rum Breath by drinking the entire bottle, but upon finishing, he passes out, drunk.