I just wanted to follow-up on my previous post so that people can have a reference for the future if they need it.
A passport was indeed all I needed to show for winning an enpou ticket. When I went up to the information counter to buy my ticket, the girl working there did ask for an id and looked a bit confused for a second. The guy sitting next to her said that foreigners only needed a passport, so I assume that she was new there and was wondering if she needed to ask me for a postal item like mentioned earlier.
Also, for the lining up process, I didn't realize at first that some of the lines were not in the main lobby area in front of the theater entrance doors. They had placed some of them in the hallway leading to the information booth and that is where enpou ticketing was lined up. It took me a while to find it as I was wandering around the crowd in the lobby area looking for my number range and couldn't find it until I asked a staff member.
It was definitely a very fun experience.