1982 240d reported to have bad injector.
 

I'm considering buying it today. What is the quickest and simplest way to determine if this is just a problem with the injector?

Take a 17mm wrench with you. While the car is running, crack open each hard injector line one at a time. This will prevent that injector from injecting fuel. With a normal injector the idle will get significantly worse when you open the line. If you find one that doesn't worsen the idle as much as the others or the nailing noise goes away, then you've found your culprit. Make sure to tighten the hard line before going to the next one. You'll probably also want some rags to clean up afterwards.

Thank you sir.

Also listen for nailing. It's a harsh sound usually on a cold engine that sounds like hammers being smacked together, or marbles bouncing around in the cylinders. What it is, is a bad injector that is popping open too soon and causing pre-detonation which is the nailing sound. This can be fixed by having the injectors tested for pop pressure, and may be solved by re-shimming it to specs. But it may also be that the whole nozzle needs replacing. Often nailing goes away once the engine warms up, but in bad cases it won't.

Let us know how it goes.

Could be other things on that cylinder as well. Best to take a spare injector and substitute it on the suspect cylinder. Then if it is just the injector you know for sure.

Otherwise a bit of a crap shoot in my opinion. Or swap locations of the same injector to see if the problem moves with the injector.

Just loosening the injector line and seeing no difference does not prove it is the injector. It could just as easily be a valve or an issue with the injection pump for that cylinder.

Sort of like Barry said but using the Injectors on your Engine.
What Biodiesel300DT said identifies the Cylinder with the problem.

One of the ways to determine if it is the Injector or not is to swap the suspect Injector with one of the other Injectors on the Engine.

Then do as Boidiesel300DT said on those 2 Injectors. If the Problem moved to the Cylinder the Suspect Injector was moved to you have an Injector Problem.

If the Problem stayed in the same Cylinder even though it has a different Injector you have a problem in the Engine itself.

You do not need to replace the Injector Heat Shields during the Test. But, when you have resolved the poroblem replace the 2 Heat Shields.