Heat 1 cup water and ¼ cup sugar to simmer in a small saucepan over medium heat, stirring occasionally until sugar dissolves. Remove from heat; add ½ cup rosemary leaves. Steep for 5 minutes then strain through a fine-mesh sieve, pressing on rosemary. Discard rosemary and let syrup cool.
Add diced rhubarb, 1 tablespoon lemon juice, 1 ¼ cups water, and ¼ cup sugar to a blender. Blend into a coarse puree. Strain through a fine-mesh strainer into a medium bowl, using a spoon to coax liquid through mesh. (You may need to do this in batches depending on the size of your sieve.) Squeeze pulp with your hands to release as much liquid as possible. Discard pulp.
Cover and refrigerate rhubarb juice and rosemary syrup separately for at least 4 hours and up to 2 days.
When you're ready to make cocktails, mix rhubarb juice, rosemary syrup, remaining lemon juice, and rum in a pitcher. Stir to combine.
Serve Rhubarb Cocktails in glasses filled with ice. Or pour into an ice filled cocktail shaker and stir until very cold to serve up in martini glasses. Garnish with a fresh rosemary sprig and/or a fresh rhubarb ribbon threaded onto a cocktail pick.