Put vodka and chopped cucumber in a nonreactive container. Refrigerate 1 hour. Strain; discard cucumber. Refrigerate, covered, until ready to use, up to 2 days.
Using a vegetable peeler, shave the remaining cucumber lengthwise into wide strips; discard seeds.
Fill eight tumblers with ice. Add 3 cucumber strips and 3 tablespoons infused vodka to each glass. Squeeze, then drop, a lime wedge into each. Top with sparkling water; stir in 1 to 2 tablespoons syrup.