Why is my left side carotid artery painful and popping during air travel?

I wouldn't expect a carotid artery to be the cause of a popping sensation at any time. Since your discomfort seems to only occur while flying, I would ascribe it to muscle or ligament changes secondary to posture or positional factors while sitting in the plane seat. Would cold air from the overhead jet be blowing on your neck? If you have significant pathology in your neck,I'd expect moresymptoms

I have never heard of a carotid artery popping and cannot think of how this might happen. Show and discuss with your team. Try to make recordings on your phone.

During takeoff and upon reaching a high enough altitude, many persons report the sensation of their "ears popping". this is actually not their ears, but their Eustacian tubes which run from the ear, behind the facial bones to the back of the nasopharynx. Popping in this area may be mistaken for the carotid artery as it originates behind the angle of the jaw where it meets the ear.

As has been well described, your symptoms have nothing to do with the carotid artery and are indeed due to eustachian tube pressure alterations during altitude changes, and merely represent a response to varying atmospheric pressures.