How should I store the cup after my period?

You can storage your menstrual cup in the pouch that came with the cup.

If you don't have the pouch make sure you keep the cup in breathable place, not in a plastic bag or closed plastic container. If you've lost your Lunette satin pouch you purchase a new one here: https://www.lunette.com/collections/accessories/products/lunette-satin-pouch