Sprinkle gelatin over water in a small bowl; let stand 5 minutes.
Combine Aperol, sugar, lemon juice, and enough water to equal 11/3 cups; bring to a simmer in a small saucepan, stirring just until sugar dissolves. Stir in gelatin mixture; heat over medium-low heat just until gelatin dissolves. Cool slightly; pour about 1/4 cup gelatin mixture over each serving of pudding. Chill until set.