Combine water and sugar in a saucepot. Heat while stirring until sugar completely dissolves. Add ginger and tea bags, and let steep over low heat (don't boil) for about 10 minutes. Cool and refrigerate. (Syrup can be used to make non-alcoholic cocktails as well. Combine with fresh citrus and some sparkling water for a fresh, unique soda.)
Combine remaining ingredients in a cocktail shaker along with 1/2 ounce of the green tea-ginger syrup. Add ice, shake well and strain through a fine strainer into a cocktail coupe (martini glass).