Cut 2 pieces of cardboard into circles that are 3 1/2" in diameter. Hot glue them together to make them stronger, avoiding the center for when you drill later. Measure and draw 2 perpendicular lines intersecting in the center. Drill through that center mark with a piece of wood behind the cardboard for safety.
Using the grid as your guide, hot glue your dowels onto the back of the clock. Using a roll of tape can stabilize the surface area for this step. A bird's eye view can help you see if the dowels align.