Are headaches after undergoing a anterior cervical discectomy plus spinal fusion significant or common?

Yes: There are multiple mechanisms by which cervical spine problems produce headaches. Nevertheless, headaches are a common symptom after cervical surgery.

They can be: Headaches are tricky, sometimes they are primarily caused by cervical spine problems, often times they are exacerbated by them, but it is difficult to predict whether surgery will ever help them, except in the case of an obvious c34 disc and radiculopathy with occipital nerve type pain which is mechanical.