However, if you play the flute too early or too late, you won't get a good pic. Play it too early and you'll just see Artis back. Play it too late and he'll be too far away. I don't own Snap so I just tried it a few times with a game of my friend and it seems that playing the flute if you are NEXT to the egg is the best time.
But you still have to take a good photo. If you play Snap, you'll discover soon that Oak isn't...well...he has a strange opinion sometimes. A cool dancing Articuno gives very little points. If you take a photo when the egg shatters and all the ice is around him, it might be better, but most times Oak tells you that it's too distorting. And if you take the photo when Arti is flying away, the size is a big problem. So try to make a photo when Arti spreads his wings and you have full look at him. This might be the best, but try for yourself!
Now, there should be another special thing to do. Remember the little Pikachu, caught by the Zubat? Use your Pester Balls and kick the stupid bat Pok�mon. However, this needs TONS of luck. But if you manage it, Pikachu will "evolve" to baloon Pikachu. Take a photo of it! Now, it should be possible that Pikachu can jump on Artis back. I never managed this, so Annarti was nice to me again and allowed me to use her info, too. She says that you have to play the flute as soon as Pikachu touchs the ground. This is NOT when you are next to the egg, but ignore this fact. Then, Pikachu will run the the hatching egg and jump on Artis back, flying away with him. Well, at least it is suppost to do it. The problem I found out: Pikachu hates the flute and it will zap poor Arti when it is next to him and the fulte is still playing. But even if the flute stopps at the right time, it is just standing next to Arti...well, maybe it's just the German version. Check out Annartis site for this, it may help you more.