Aerosol cans, razor blades and inexpensive tweezers are among the biggest rust stain culprits in your bathroom. Keep metal items such as these away from direct contact with your counter, if possible, either by storing them inside a cabinet or placing them on a towel-lined tray on the counter. Don't use metal items such as toothbrush holders or soap dishes on your countertop, either; instead, use only items made of rustproof materials such as porcelain or plastic.
Some bathroom countertop rust stains are going to require chemical cleaning, but if you catch it soon enough, it's possible to remove it naturally with vinegar and salt. Pour a small amount of white vinegar on the rust stain and scrub it with a nylon pad. Sprinkle salt on the stain and let it sit for an hour, then scrub it again. Rinse well. Do not use vinegar on stone countertops, as the acid may etch the surface.
Mild, abrasive-free powders such as Bar Keeper's Friend and Bon Ami, found near the abrasive scouring cleansers in the supermarket, remove moderate rust stains on countertops. Simply moisten the stain with a damp sponge, sprinkle the cleaning powder on top and scrub. If the sponge isn't strong enough, use a nylon pad or a toothbrush. Once the stain is gone, rinse thoroughly with water and wipe dry with a microfiber cloth.
If the rust stain on your bathroom countertop is old and doesn't budge or disappear with other methods, it's time to try something harder. Industrial cleaners for rust, lime and calcium are available commercially in supermarkets and home centers. Ventilate the bathroom by opening the door and windows and turning on the exhaust fan before using a chemical cleaner, and wear protective gloves. Dilute the rust remover with water as directed and apply it to the stain with a sponge or toothbrush, or use it full-strength and rinse immediately with water.