Bring granulated sugar and 1 cup water to a boil in a medium pan over high heat. Stir until sugar is dissolved. Refrigerate until chilled, about 1 hour.
Add lemon juice, superfine sugar, cranberry juice, and 1 1/2 quarts water to a serving container. Stir until sugar is dissolved. Refrigerate until ready to serve, up to 1 day.
To serve, add ice cubes to fill and garnish with lemon slices.