Can I get multiple effects from Alhammarrett, High Arbiter when using Brago, King Eternal's ability?
The effect disappears as soon as you blink the Arbiter.

400.7. An object that moves from one zone to another becomes a new object with no memory of, or relation to, its previous existence.

The rules list a few exceptions, but the situation you describe isn't one of them. The blinked Arbiter does not know anything about the Shock you chose when it was cast.