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The title is a play on the phrases "castaway" and "Anchors Aweigh," alluding to the nautical theme.
Sylvester brings his son to the wharf to scavenge for fish heads. Junior laments the fact they have to scrounge for food and wishes they had a real home. When they see a ship advertising for a mouser, Junior urges his dad to apply. Sylvester doesn't want to work, but agrees to take the job when Junior says he'll help.
Sylvester lounges aboard ship while his son chases the mice. When Junior bemoans the iniquity, Sylvester says Junior is a little cat so he should handle all the little mice, and that he himself is a big cat so he'll take care of any big mice that come along. Soon Junior chases a little mouse into a crate in the cargo hold. He opens the crate to get the mouse, but out hops Hippety Hopper, a baby kangaroo. The cats think it's a giant mouse, so Sylvester tries to hold up his end of the bargain, but Hippety pummels him.
Sylvester now strikes a new bargain. Since there are a whole bunch of little mice, he should take care of them, and Junior can take care of the big one. Junior agrees halfheartedly and soon encounters Hippety again. He notices that Hippety imitates his actions, so he starts dancing and leads the kangaroo back into its cage. He proudly shows it to his father, who again tries to uphold his end of the bargain. Sylvester chases a little mouse around the hold until it scurries into a crate of Super Muscle-Building Vitamins...
The version of this cartoon shown on CBS cut the part where Sylvester chases a mouse into a crate of Super-Muscle Building Vitamins and the mouse ingests them and becomes super strong, making the ending where Sylvester gets pummeled by the mouse inexplicable.