Put the celery, 1 Tbsp of the sugar, celery seed and salt in a food processor. Pulse until the mixture is reduced to a puree. Pass it through a fine-mesh strainer, pressing on and discarding the solids.
Place the strained celery juice in a saucepan, along with the 1/2 cup sugar. Heat gently until it just dissolves, then cool
To assemble the juleps, shake together 2 ounces whiskey, 1 Tbsp lemon juice, and 2 Tbsp celery syrup with ice. Pour into a glass with ice, top off with seltzer, and garnish with a celery sprig. Adjust to taste.