In a pot, combine the sugar, water, and lime zest. Place over medium heat and bring to a simmer, stirring constantly, until the sugar is dissolved. Remove from the heat and stir in the mint leaves. Let the mixture sit for 30 minutes.
Strain the syrup through a fine mesh sieve. Combine the syrup and the lime juice, then add in about 3/4 of the cucumber, reserving the rest for garnish. Refrigerate for at least one hour, but overnight is best.
When ready to serve, pour 0.5 cup of the sparkling water into a cup. Add 6 tablespoons of the cucumber lime syrup. Serve.
Notes
Garnish with reserved cucumber slices and a mint sprig for an extra touch.