Both A and B are correct but B is an older form and is less commonly used now - you are more likely to see it in older literature and more likely to hear it in the negative form in phrases like "You needn't worry" "Well you needn't think I'm going to do it for you!"
No "Need you a new hat?" is an archaic form - you can't say it now (other than perhaps in a few strong local dialects).
"Need you [do something]?" still occurs as in your example "Need you go?" but never "Need you [a thing]?"
Generally I would say don't write or use the B form in speech but understand it for the occasions you will come across it, mainly in literature or period drama.
Hope that helps :)