Rinse the cranberries in a strainer under running water and put them into the blender.
Add the remaining ingredients and process on high speed for 1 minute or until the cranberries are completely broken down and the nog becomes creamy and thick. There may still be tiny flecks of cranberries visible even after blending thoroughly.
Pour the nog into a pitcher and chill until ready to serve. Covered and refrigerated, this nog will keep for 2–3 days.
To serve, fill small punch cups with nog, sprinkle each serving with a dash of nutmeg, and stand a cinnamon stick into each cup.