The Buzzer itself is part of the cluster. if its the buzzer then its (unless you are a master craftsman with it comes to soldering) a replacement cluster thats needed.
Completely cost prohibitive in my opinion.
As for testing for errors, yes you can throw VCDS on the car, but i dont remember ever getting an error on the computer for this problem (ive had 2 different cars that suffered with the same thing... an A4 and an A3).
What i might suggest before you go ripping into the dash to look at the rear of the cluster, is checking other connections around the dash. particularly around the light switch or the indicator stalk.