Hi,
Going around the Tokyo (Kanto) area for day trips is fine. Although Nikko is better with an overnight.
For many places being closed, of course the public parks and stations are open. But almost everything with a door or gate will be closed. Museums, department stores, banks, castles, many restaurants, etc. Some temples, shrines and convenience stores are open for New Year's though. Before going someplace during that period, I suggest you look at the website of the place and make sure it is open beforehand.
Since you'll be in Japan for New Year's, I suggest you see how Japanese celebrate it. There is no grand party in the streets with champagne singing Auld Lang Syne, but at many shrines and temples on New Year's Eve and New Year's Day there is a wave of people going to visit, called Hatsumōde. In Osaka one very busy place is Sumiyoshi Taisha, and in Kyoto there is the Fushimi Inari Shrine. You might see a very long line of people waiting to go say a prayer for the new year.
Good luck.