Hello, I'm Jason. I will do my best to answer your question quickly and accurately using the information you have provided.
I will need to know what codes are stored in the engine control module to be sure about this diagnosis
if it's just misfire codes then it's not likely that injectors need to be replaced
Here is the text of the Audi service bulletin that I'm referencing:
Sporadically, the engine may run slightly rough for a few seconds after cold start. The MIL may illuminate with a DTC for misfire on one cylinder only (P0301, P0302, P0303, P0304, P0305 or P0306) stored in the ECM data memory. The rough running condition disappears a few seconds after cold start and does not exist with a warm engine.
Ensure that TPL(NNN) NNN-NNNN- Cold engine runs rough or stumbles; warm engine may stall when engaging gear and vehicle is stationary has been performed.
Now, there is another TSB stating the following:
One or more fuel injectors may be stuck open, allowing fuel to enter the combustion chamber when engine is hot. Condition may cause the air fuel mixture to be too rich, causing a delayed restart of the engine or "long crank."
1. Bring engine to operating temperature and continue to hold the engine at 3,000 RPM for an additional 5 minutes, so that the engine is hot.
2. Record MVB 140 to determine rail pressure. Should be at ~35 bar.
3. Turn engine off.
4. Monitor MVB 140.
If the pressure decreases to 25 bar within 10 minutes, remove the plugs and check to determine which injector is leaking fuel into the cylinder.
5. Replace leaking injector(s).
I would definitely want to know exactly what steps they have taken before proceeding with the repair
In addition, if it does need new injectors, the 1900 price seems quite high, since fuel injectors cost $135 each. Seems like very high labor