Lay the curtain that you need to repair out on a flat work surface where you can examine the backing thoroughly. Find all of the spots that need to be repaired.
If there are any tears in the backing that can have the edges sewn together, do that now so the surface will be flat and uniform before a patch is applied. When sewing these edges together, do not allow the thread to go through the front fabric of the curtain.
Cut the patching fabric into appropriately sized pieces for covering each tear or hole in the curtain backing. Cover the damage, especially any holes where light can shine through, with a fabric patch.
Sew the patches on by hand, using the thimble to press the needle through the two layers of blackout fabric if they are especially thick. Do not pierce through the front of the curtain while sewing.
Hold the finished curtain up to the light to make sure that the new patches block out light. If you notice any light coming through the curtain, repair that spot again with a new, possibly larger, patch.