For anyone looking to get this done easily, go to the contracts section at the title menu, select the "Featured Contracts" option and sift through all of them until you find the one called "Mr. Nice Guy". I found this on the 3rd page.
This one takes place right at the default beginning (Red Carpet) of the Showstopper mission, and the guy you need to take out is right behind you at the start. I took the syringe, injected him in the first 3 seconds of the game, and ran to the exit (which was two feet away
). Mission over in ~10 seconds.
I did get spotted, and people were shooting at me, but you can still exit the mission easily before anyone has a chance to put you down.
Questions, corrections and neg vote justifications below.