If I cast the arbiter, chose a spell (let's say Shock) and then blink it via Brago's ability, I know I can choose another spell (maybe Goblin Locksmith), but can I have both Shock and Goblin Locksmith 'blocked' at the same time? Does the effect on shock disappear when the arbiter blinks, limiting to one spell 'blocked' at once?
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.