Hatsumōde (the year's first visit of a Shinto shrine, less commonly a Buddhist temple) happens not necessarily on the 1st - some people do it on the 2nd or 3rd (or already on new year's eve). Expect popular shrines such as the Meiji Shrine to be extremely crowded so that it can take an hour or more to get to the front - but many people will wear traditional clothes and it can be quite an interesting spectacle. And nobody will take offense at foreigners attending if you behave politely.
In many large Buddhist temples, there is a ceremony at midnight on new year's even where a bell is rung 108 times, intended to purge the earthly temptations that prevent people from achieving nirvana.
Yasukuni shrine holds a festival on January 1st that includes various performances of music, dance and martial arts.
Another event is the Emperor's new year's speech on January 2nd, one of only two dates where some of the inner parts of the imperial palace are open to the public, and of course seeing the emperor and his family in person is also quite exceptional.
In contrast, Christmas is not a national holiday in Japan, and there are no traditional events - unless you count recent commercial traditions. Department stores will have lavish decoration, and there are commonly family dinners or romantic dates.