Make blueberry shrub: Bring vinegar, sugar, and 1/2 cup water to a boil in a small saucepan. Stir in berries. Let cool, then chill airtight at least 4 hours and up to 1 week.
Fill 4 tall glasses one-third full with ice; arrange cucumber sticks in them. Pour 1/2 cup soju into each, followed by 1/3 cup shrub (liquid and fruit). Taste, then add more shrub and a splash of club soda if you like.
*A liquor traditionally made from rice, soju can also be distilled from wheat, barley, tapioca, and sweet potatoes. The Japanese style is generally smooth, clean, and subtle; the Korean tends to have a fuller, stronger flavor. Find it at well-stocked grocery stores and liquor stores.