I don't understand what you mean by fail. Please elaborate. I would really appreciate it. It would just be a daily driver, not trying to race or anything like that. I believe it is DSG. If you look at the pics up closer.
I'm not sure if it's DSG, but just cause it's an automatic trans, doesn't mean it's a DSG transmission. DSG is a dual-clutch, but your normal auto is a slush-box, not DSG...not necessarily the same trans found in the GTI/GLI platform. I'm not super familiar with the A3 of these years. Ask the owner if it has the DSG trans or the slush-box auto. Regardless, it's an Audi with an auto trans, NON Quattro. So no AWD. Proceed with that from there.
Based on the write up of the car much of the important care has been done saving you from inheriting a car that suffers from delayed maintenance. It does have the DSG trans just like the GTI. Its much like a GTI just a little more luxury in the interior, drivetrain is identical. Should make a great commuter car and you should be able to buy yourself a new center armrest easy enough.
One thing you should do before buying the car is have the cam follower under the fuel pump checked. It is a point of failure and can cause major engine damage. There is no mention of it in the write up but the owner may have addressed it. Part is only 49 bucks and is fairly easy to install but it is a known source of issues for this model motor (TFSI).