Only shots taken after the 1.24 patch has installed will count towards it.
Every goal you score also counts as a shot on goal so a very easy way to unlock the trophy is to just play a 1v1 exhibition game with no bots, and keep scoring off the faceoff. While your stats will not be counted in the stats screen, the game will keep track of your shots still so the trophy will still unlock even if your stats say otherwise! Also note that you must not disable Goal Resets, else only your first shot will count and the rest won't.
There isn't any difference between 5/10/20 minutes matches because the game tracks every shots you make, so the trophy will pop in the middle of a game. Remember that a 5 minutes match will last 15/20 minutes because of all the goals scored (between 100-130) so about 1.5 hours of doing this will get the trophy.