Grind rice finely using a coffee or spice grinder. Transfer to a large bowl; add cinnamon sticks. Pour in warm water. Cover with plastic wrap and let sit, 8 hours to overnight.
Strain rice mixture through a sieve set over a bowl. Press against the rice with a wooden spoon to squeeze out as much liquid as possible; discard rice. Stir rice milk and sugar into strained liquid to make horchata.
Measure 2 1/2 cups horchata into another bowl. Add cinnamon whiskey, cayenne pepper, and 1/4 teaspoon ground cinnamon. Pour into ice pop molds.
Freeze until solid, at least 12 hours.
Run molds under warm water to release ice pops. Sprinkle with an additional dash of cinnamon before serving.