I couldn't find fig puree, so I made my own with Trophy dried soft smyrna figs. Chop the figs and add them to a pot with water. Bring it to a boil and cook until figs are very soft. Let cool. Puree with hand blender or magic bullet (make sure that the mixture is not hot). Add lemon juice to thin down if necessary.

You don't need to make your own simple syrup. Starbucks sells it as Classic Syrup as do other brand that sell coffee syrups.