Combine sugar and 1/4 cup water in a microwave-safe dish; microwave at HIGH for 2 minutes. Stir until sugar dissolves; cool.
To prepare cocktail, place cucumber piece in a martini shaker; muddle. Add Suze, juice, and 1 1/2 tablespoons syrup; reserve the remaining syrup for another use. Cover and shake. Strain mixture into a highball glass with ice. Top with beer; garnish with cucumber slice.