Combine the sugar and 1/2 cup water in a small saucepan and bring to a boil, stirring until the sugar is dissolved. Lower the heat and simmer for 5 minutes. Turn off heat and let cool to room temperature.
In a rocks glass, muddle 4 of the blackberries, 1/2 mint sprig and 1 ounce of the cooled simple syrup. Add 2 ounces of the vodka and 3 ounces of the blackberry juice, and fill with crushed ice. Stir gently. Top off with 2 ounces of the club soda. Garnish with a blackberry and a mint sprig.