With needle or without?
Most syringes can attach to plastic tube (IV line) - so depending on costume, could could secrete an IV bag with a short bit of tubing running to the ear - provided the actor can use two hands (one to hid the action whilst assisting with the connection). That is one option.
With a needle - I would suggest palming a milk creamer - contents replaced with your blood substitute and re-sealed with tape. Actor 'restrains' the head with one hand (with the creamer palmed), plunges the syringe into the top of the creamer and holds on to the base of the syringe with the hand with the creamer then draws the plunger. Just make sure the needle is not longer than the creamer (cut it down if need be)
I have always found magicians/magic shops to be a great place for solutions requiring some trickery - there is probably an off the shelf syringe that can do something similar to this trick too.