In a saucepan, mix together the sugar and water over medium high heat and bring to a boil. Stir to dissolve the sugar. Turn the heat down to low and add the strawberries. Cook for about 10 minutes. At the end, add the juice of 1-2 limes. Set aside to cool completely, then refrigerate until ready to use.
For the Moscow mule, muddle the mint leaves and strawberry-lime simple syrup in the bottom of a glass to release the flavors of the mint. Add ice, and then pour in the vodka, lime juice, and top with the ginger beer. Stir well and serve immediately.