If it's truly rust, phosphoric acid will remove rust from stainless steel. If you have any Ospho metal primer, you can try that. Or dump a bottle of Coca-Cola in it.
If it's actually coffee crud or some other beverage stain, washing soda (calcium carbonate) is very effective at removing coffee deposits.
I would start by dumping a couple tablespoons of washing soda in it, filling with hot water and allow it to sit overnight. Scrub it out with a bottle brush (if you have one) and rinse. If it's still rusty, hit it with phosphoric acid. If it is good quality 300 series stainless, any rust should be only a thin film.