For the ginger syrup: Add the sugar, ginger and 1/2 cup water to a small saucepan. Bring to a boil over medium heat, stirring to dissolve the sugar. Remove from the heat and set aside to cool completely. Strain out the ginger.
For the cocktail: Stir together the bourbon, kombucha, and 1/2 ounce Ginger Syrup in a rocks glass filled with ice. (Reserve the remaining Ginger Syrup for additional drinks.) Top off with the soda water. Garnish with mint and serve.