Does getting your ear pierced and the flu shot hurt? should i do it on different days?

No: I am not aware of any reason you can't do both on the same day.

Yes/no: Both issues will involve brief insults to the skin integrity and trigger pain sensations. Both are very minor events to the body. Unless you faint easily or have a phobia to minor pain/needles, i see no reason to spread them out over several days. Even then, the dread would be worse than having it done.