I've had this little popped vein in my eye for as long as I can remeber, how can I get rid of it?

Irritation. You may have blepharitis, dry eye syndrome, allergies, or something causing a problem in blood flow like a cc fistula. This should be looked at by your ophthalmologist.
Lubrication. Sometimes conjunctival veins can be accentuated from dry eyes which can be reduced with treatment of dry eyes. However, to remove it completely would require surgery. It can be done but its considered cosmetic and not really recommended.