Combine sugar and 2 tablespoons hot water in a large jar, cover, and shake until sugar is dissolved. Add gin, cilantro, and lime juice (save gin bottle for finished product). Cover and chill 2 days. Strain into a medium bowl; discard cilantro. Pour cilantro-lime gin back into reserved bottle.
Assembly
For each cocktail, pour 3 ounces cilantro-lime gin into a cocktail shaker filled with ice. Cover; shake until cocktail shaker is frosty, about 30 seconds. Strain into a chilled Martini glass and garnish with lime.
DO AHEAD: Cilantro-Lime Gin can be made 1 month ahead. Keep chilled.