How many days will it take for a metal spoon to rust in a cup of salt water?

Not all metal spoons are made of exactly the same quality of stainless steel. Do not assume manufacturing uniformity. Furthermore, not all cups of salt water have the same concentration of salt (although if you had specified sea water, then we would have a way to know approximately how salty the water is - even that is not 100% uniform, the sea is not exactly the same everywhere in the world, but it is reasonably close). So there are two unknown variables in the process you describe.

A good quality of stainless steel is remarkably resistant to rust, even when it is immersed in salt water. It could take a really long time.