Do ahead:Over medium heat, add sugar and water in a small pot. Bring to a boil. Add the 1 cup mint leaves and boil 1 minute. Remove from heat and let sit 30 minutes to infuse. Strain infused syrup into a container and set aside. (Any leftover syrup can be stored in the fridge for future use.)
Cut the lime into quarters.
Squeeze the juice of two lime quarters into a highball glass, adding the squeezed lime into the glass as well.
Add half of the mint leaves and cherries into the glass and muddle together, taking care to only bruise the mint leaves and not crush them.
Fill each glass with ice to the top. Add 3 oz white rum and 2 oz of the infused syrup to each glass. Top each glass with club soda, gently stir, and serve immediately.