Combine cranberries, sugar and water in a saucepan over medium heat. Bring to boil and then reduce heat to simmer.
Simmer for 10-15 minutes; stirring occasionally. Remove from heat and let cool. Strain liquid and discard berries.
For each cocktail: Add 7-10 mint leaves and 1 Tbsp lime juice to a glass. Muddle mint.
Add 2 oz cranberry syrup and 2 oz of rum to each glass. Top with 2 oz prosecco. Garnish with additional fresh cranberries and mint if desired.