Use the ladder to reach the ceiling crack. With a utility knife, remove any loose material that might be inside or around the crack. Abrade the painted surface around the crack a bit with sandpaper. Wipe the whole spot with a damp rag to remove any dust or debris.
Check the area around the crack to determine whether or not that part of the drywall ceiling is firmly attached. If it's not, secure it further by driving drywall screws into the framing on each side of the crack.
With a drywall knife, apply a coat of drywall joint compound to the crack. Place a strip of paper drywall tape across the crack, embedding part of the tape into the wet joint compound. Let the patch dry.
Apply additional coats of the joint compound to the taped crack using the larger drywall knife. Spread this top compound layer flat across the whole crack and its immediate surrounding spots, to the same level as the rest of the ceiling. Let the compound dry.
When the top compound layer is dry, lightly sand down the whole spot where the crack was until it’s smooth. Use the damp rag to clean its surface of fine dust.
Apply drywall primer to the whole repaired spot and adjacent portions. Let the primer dry, then repaint the area with wall paint of your choice, preferably of the same hue as the rest of your ceiling.