Your arm will heal in time regardless of what you do,.. but these things will help the pain immediately, and speed recovery time..
Expect your arm to be stiff (especially at the inner elbow joint) for a day or two... With tissue trauma this is perfectly normal (though it shouldn't happen with a routine blood draw).
The only time you need to seek medical help is if your pain becomes worse AND you become feverish (measurable). If this happens, then it would be a good idea to go ahead and see a doctor just to be safe. This shouldn't happen though.. in 99% of cases of botched blood draws the patient just has a day or two of pain and then they wake up on day 2 or 3 and the arm feels much better.
Though again, I would expect significant bruising to appear soon. If not, great.. but a deeper puncture means the blood will take longer to reach the surface of the skin.
You do not need to be worried -- just use Ice/Heat (if you want) and Ibuprofen.
Take care,