Christmas is not a Japanese national holiday. Businesses and schools are open unless it falls on a weekend. For the reason, traveling around Christmas Day in Japan isn't as bad as in Western countries. One thing to remember is that Christmas Eve has become a night for couples to spend a romantic time together at fancy restaurants or hotels in Japan. If you plan to go out on Christmas Eve, it's recommended to make reservations as early as possible.
New Year's holidays are very important for the Japanese, and people usually spend New Year's Eve rather quietly with the family. As many people travel out of Tokyo to visit their home towns or go on vacation, Tokyo is quieter than usual, except temples, shrines, store sales, and so on. It might be a good idea to stay in Tokyo during this time. You may get good deals for nice hotels. On the other hand, onsen hot springs and snow resorts tend to be crowded with visitors. Early reservations are recommended if you plan to stay at onsen or snow sports destinations.
Most of businesses in Japan, including medical institutions, are typically closed around the 29th or 30th of December to 3rd or 4th of January, depending on the kind of business and day of week. In recent years, many restaurants, convenience stores, supermarkets, and department stores are open even during New Year's holidays.