Combine the sugar, 1 cup water, ginger and grapefruit zest in a small saucepan; bring to a boil over high heat, stirring until the sugar dissolves. Remove from the heat and add most of the mint leaves. Let cool to room temperature, then chill for at least 4 hours and up to 24 hours.
Strain the simple syrup into a small bowl and discard the solids. Divide the grapefruit segments, lime pieces and the remaining mint leaves between two tall glasses. Add a few tablespoons of simple syrup to each and muddle. Fill the glasses with ice cubes; stir 2 tablespoons of the rum into each and fill with sparkling wine. Stir, garnish with mint sprigs and serve.