In a medium bowl, stir together the rice, cinnamon, and 2 cups boiling water. Cover with plastic wrap and set aside at room temperature for at least 8 and up to 24 hours.
Transfer the rice mixture to a blender. Add the coconut water and blend until smooth, 3–4 minutes. Line a sieve with cheesecloth, and strain the rice mixture into a pitcher, pressing on the solids to extract as much liquid as possible. Discard the solids; cover the liquid with plastic wrap and refrigerate until ready to serve.
In a small pot set over low heat, stir together the coconut milk, sugar, and salt; cook, stirring occasionally, until the sugar dissolves, about 3 minutes. Remove from the heat and set aside to cool to room temperature.
To serve, stir the cooled coconut milk mixture into the chilled rice mixture, then pour the horchata into ice-filled glasses. Grate a little bit of fresh cinnamon over the top and drop the sticks into glasses to garnish.