In a large pot, stir together the cooked rice with the sugar and all but 1/4 cup coconut milk. Bring to a simmer, reduce the heat to low, and cook, covered, stirring occasionally, until the rice is very soft and the milk is thick, about 30 minutes.
While the rice pudding is cooking, place the white chocolate in a medium glass bowl and place over a small pot of simmering water to melt, making sure to stir frequently. You can also melt the chocolate in the microwave at 15 second intervals until melted and smooth. Place the colored sugar into a small bowl. Dip half of each fortune cookie in the white chocolate then into the colored sugar and place on a wax paper-lined sheet tray. Allow the chocolate to set in a cool place for about 30 minutes.
To serve, scoop the rice pudding into serving glasses, drizzle with the remainder of the coconut milk, and serve with an embellished fortune cookie.