To make the syrup, put the rhubarb, sugar and 1 1/4 cups water in a medium pot. Bring to a boil over medium heat, stirring occasionally until sugar dissolves. Reduce the heat to a simmer and cook until rhubarb is soft and falling apart, 20 to 25 minutes.
Strain through a fine-mesh sieve and cool the syrup before using. You can either discard the rhubarb or save for mixing into plain yogurt or spreading on toast. Syrup will keep covered in the refrigerator for up to 3 weeks.
To make the drink, combine all the ingredients in a cocktail shaker with ice. Shake vigorously until the outside of the shaker becomes frosty. Strain into a chilled glass.