Hi. Japan’s winter season is December through February. Depending on what part of Japan you are travelling to, winter in Japan can be vastly different. Big cities such as Tokyo and Osaka generally experience short winters with cold but sunny days and occasional snowfall. But the further you head into the mountainous regions and Northern parts of Japan the more the temperature starts to drop and the winters can be long and cold with high levels of snowfall for many months of the year.
The large amounts of snow that fall each year makes conditions perfect for skiing, snowboarding and other winter sports. Winter festivals, such as the Sapporo Snow Festival held in Hokkaido, are a must-see. Various snow and illumination events are held across the country during the winter season creating breath taking views in this winter wonderland which draws just as many tourists as during the warmer months.
Moreover, Japan is notoriously crowded, and even more so during the mild months. The spring sakura and autumn koyo attract millions of tourists each year, while the summer holidays for many other nations means kids and families have more free time to travel. By visiting in the winter, travelers can experience a totally different side of Japan and avoid the madding crowd. As such, hotels will be more economical and available to tourists.
However, many tourist attractions, stores, and restaurants are closed on one or more days between December 29 and January 4, limiting your sightseeing, shopping and dining choices, especially on January 1.
Museums are typically closed for multiple days over the holiday season. No nationwide pattern can be recognized for gardens and castles, some of which close on multiple days, others on a single day and others not at all. Temples and shrines naturally do not close over New Year.
Shops typically close on January 1, but are open on all other days around New Year. In recent years, an increasing number of shops remain open even on January 1, especially in modern shopping districts and malls of larger cities.
Restaurants typically close on one or more days over the holidays, especially January 1. Many fast food chains, hotel restaurants, as well as the restaurants in the above mentioned modern shopping districts and malls, also open on January 1.
So your best bet would be to visit popular shrines and temples where you can experience a festive atmosphere with food stands and many people lining up for a prayer at the main hall, purchasing lucky charms for a fortunate new year and disposing their lucky charms of the past year. Most atmospheric visit to a temple around midnight on New Year's eve, when the temple's bell is rung repeatedly.
I am not sure if hotels are in hight season, but it is indeed one of the travel peak in Japan as a lot of people will travel to see their family.
Not everything is close, but indeed there will be some restaurants, stores and venues that will be close for the new year. I do not think that this should stop you from going at this time of the year.
As new year is an important celebration in Japan, it's a good chance to see some interesting aspect of the Japanese culture. If I am correct, new year eve is the only night in the year that the train run all night in Tokyo. You might want to go to a temple at midnight, they are rigning bell and people go to pray for good luck on the first day of the year. If you go to a big temple like Senso-ji, you are sure to find a huge crowd.
I wrote an article on pilgrimage and one thing that you might want to try in the first days of the new year is a course to visit 7 temples and shrines dedicated to the seven gods of luck, it is called "shichifukujin meguri", there is many in Tokyo, I personally did the Yanaka course that go between Ueno an Tabata stations on the Yamanote line.