Wrap cardamom in cheesecloth and tie with kitchen twine into a tight packet; set aside.
Combine sugar and water in a 1-qt. saucepan over medium-high. Cook, stirring occasionally, until sugar is dissolved, 2-3 minutes. Remove from heat and add cheesecloth packet. Let syrup cool completely and chill 8-10 hours.
Remove packet and refrigerate syrup until ready to use.
Cocktail:
Combine tequila, syrup, and juices in an ice-filled shaker. Shake well.
Strain into a rocks glass with 1 large cube; garnish with lime slice.