In a small saucepan, bring 1 cup water and the sugar to a boil over medium-high heat, stirring until the sugar has dissolved. Simmer, stirring occasionally, until the mixture is reduced by half, about 15 minutes.
Remove the saucepan from the heat and stir in 1/2 cup mint leaves and the bourbon. Cover and let steep for 1 hour.
Place the cantaloupes in a serving bowl. Strain the syrup over the melon; stir in the lemon juice to taste. Cover and refrigerate for 1 hour. Top with mint sprigs.