If you have neck pain over your carotid, do you need an ultrasound?

Not necessarily. Check with your physician but pain over the carotid may arise from your cervical spine. But a detailed examination is paramount and an ultrasound may be useful.
Possibly. Typical symptoms of carotid disease are stroke, dizziness, sudden loss of vision in one eye. It is not generally painful. Other problems in the neck like tumors or trauma can involve the carotid and have pain. Most neck pain, however, is muscle or nerve related and has nothing to do with carotid artery. Only a health professional can guide you after examining you.