Sudden bulging of left temple vein that doesn't go away - Is it treatable? (Photo)

First it needs to be clear from an examination if this is an artery or a vein. You need an in person examination to confirm this is a vein, you are concerned with.  If it is a vein, which is likely by its position treatment can be done to remove it.

A temple vein can be safely treated by sclerotherapy ( injection of a sclerosant solution, actually foam). 
A phlebectomy seems possible too. 
A temple artery would be pulsating, a simple touch will tell you the difference. 
See a vascular physician in your area if it is a vein.

Thank you for the question! It is rather difficult to fully assess your situation without an in-person exam. Such an exam would be necessary to ensure that your bulging vein is not an artery. If you do have a varicose vein, however, it can be effectively treated with sclerotherapy. This method uses a sclerosing solution that causes the vein to shrink. Usually sclerotherapy takes only 1 to 3 sessions to achieve fading of the undesirable veins.  
An in-person exam with a board-certified plastic surgeon is the best way to assess your needs and obtain expert medical advice. Best of luck!