Can you get brain damage from lack of oxygen fainting?

Yes: The human brain can withstand at most 4 minutes of oxygen deprivation before experiencing brain damage. Damage may be irreversible at this time but the brainstem is more resistant, hence the occurrence of "persistent vegetative state". But a simple faint can result from only a few seconds of oxygen deprivation without causing long-term damage.

Generally no: A simple faint restores adequate blood flow to the brain as you become horizontal, and recovery is forthcoming. If you have brain damage from a faint it can be from a near- fatal arrhythmia (such as from long qt syndrome or hypertrophic cardiomyopathy.), or prolonged seizure. A physician can help determine if there is cause for concern.