Purchase a pumice stone in the cosmetics section of a drugstore.
Moisten the end of the pumice stone.
Rub it against the rust, like you were erasing it with a pencil eraser. The rust will come right off.
Rub the rust roughly with a moistened scrub pad. That alone may be enough to get the stain out safely without scratching your surface. If not, go on to the next step.
Dry the sink drain.
Moisten a few paper towels with Coca-Cola.
Layer them on top of the rust stain and leave them overnight.
Wipe what remains of the stain the next morning with a dry paper towel.