How long would you need to stay in the hospital after a small stab wound near the neck?

Depends. If there is no major damage to nerves, blood supply, organs then early if there are complication with breathing or eating or any other issue will need a transitional care hospital.
Depends where. There are several different areas in the neck with different degrees of danger regarding structures that could be damaged. Usually the higher up in the neck te more dangerous, but lower down it can also be possible to damage the lung. It all depends on what was damaged. Usually the larger the knife the more dangerous too, but a very small puncture wound can sometimes also be very dangerous.