Using the pen tool, draw a mask (onto the rainbow layer) that is arch-shaped. You only need two points for this. Next, apply effect>render>stroke to the rainbow. Adjust the brush size until it’s nice and fat for now. Next change the effect’s paint style to “reveal original image”. Adjust the position of the mask and the brush width until the entire rainbow can be seen.
Now to reveal the rainbow, use keyframes to animate the “end point” value of the stroke from 0 to 100.
If you want to reveal each band of colour separately, you can create six coloured solids, each with a mask and stroke as above, then animate their reveals separately.
Hope that helps,
Steve