Sand the rusted areas in the cast-iron sink gently with a 000-grade sandpaper three to four times. Brush the loosened rust particles aside with a rag as you work so you can inspect the areas. Continue sanding, removing all the rusted areas from the sink. Remove the rust particles from the sink with a hand vacuum.
Pour white vinegar into a spray bottle, filling it halfway. Fill the bottle to the top with water. Spray the cast-iron sink generously with the diluted vinegar. Allow the vinegar to remain on the sink for 30 minutes. The vinegar dissolves any rust remaining inside the chips and nicks within the sink. Scrub the sink and inside the chips and nicks with a nylon scrubber. Rinse the sink with water and a sponge.
Mix 1 part bleach with 10 parts water in a bucket. Dip a sponge in the bleach water and wipe the entire sink, disinfecting the surface. Rinse the bleach completely from the sink with water. Dry the sink with a clean cloth.