Is there a way to pick up an arduino off the bench and read what the program is uploaded on it?
Yes there is but it is very very complex and you must be able to understand a compiled hex file.
If you dont have those skills then the answer is probably NO.
@ballscrewbob already gave a good answer while I was writing this, but since my answer is slightly different (though no much), I'll post it anyway:
You can read the binary program off of it, but you can't read the sketch code off of it.
It is possible to write your sketches so they will output some identifying information (such as printing the sketch filename and timestamp of the upload to Serial Monitor), but you have to intentionally do this, it's not done automatically.