Pulse 2 tablespoons cilantro and the sugar in a mini food processor until very finely chopped; transfer to a small plate. Rub the rim of a chilled martini glass with the lime wedge, then dip in the cilantro sugar to coat.
Muddle the remaining 1/4 cup cilantro and the lime juice in a shaker with the handle of a wooden spoon. Add the vodka, Aperol and cranberry juice, fill with ice and shake well. Strain into the prepared glass.