Muddle the mint leaves with the simple syrup and lime juice in a cocktail shaker. You can use a cocktail muddler or just a spoon.
Add the rum, bitters, and ice. Shake until well-chilled, about 60 seconds.
Strain into a chilled coup serving glass.
Top with champagne. Add any desired garnish and serve immediately.
Notes
Use fresh ingredients: Fresh mint leaves and freshly squeezed lime juice are essential to creating a refreshing and flavorful cocktail. If the mint is not fresh, it can be bitter and ruin the flavor of the drink.
Muddle the mint well: Muddling the mint helps to release the essential oils and flavors. So be sure to muddle your mint for long enough to get the most out of it.
Use a good-quality rum: A good-quality rum will make a big difference in the flavor of your Old Cuban. So choose one that you will be able to enjoy best in this drink.
Get the sweetness right: The right amount of simple syrup will depend on personal preference, but it's important to add enough to balance the tartness of the lime juice. Feel free to adjust the amount you add to suit your taste.
Chill the ingredients before mixing: Chill the mint leaves, lime juice, rum, champagne, and simple syrup before making your cocktail. This will help to keep the drink cool and refreshing.
Measure accurately: Use a jigger or a measuring cup to measure your ingredients accurately. A precise measurement can make a big difference in the balance and flavor of your cocktail.