We were on a 4 day cruise over New Years in 2001. It was a great trip - we boarded on Dec 31 and there were all kinds of New Years decorations and special items about the ship. Everyone had a special dinner menu (lobster, etc, and most people wore formal clothes) in whatever restaurant they were in scheduled for that evening - with champagne supplied. After dinner there were parties in each of the nightclubs and the atrium - free champagne and hats/streamers/noisemakers all round. They had all the parties linked through video so the whole ship could participate in a giant countdown. It was a fabulous time and we danced until the wee hours. The next day was somewhat subdued, but they were showing football games in the ESPN lounge. I'm not sure what they would do differently on a 7 day cruise - but I'd be willing to wager that you will have a great time!
Enjoy your cruise,
Lorraine