In Japan (as in the US) everyone celebrates Solar New Year on January 1. This is a big deal celebration, with all the traditions, all the national holidays, all the special food.
Traditionally, new years was celebrated according to the Lunar (sometimes called Chinese) calendar, which is later in the year. In 2016, we will welcome the Year of the Monkey on February 8.
Many traditional references to new year's celebrations will talk about the "end of winter" or the "start of spring." This is why -- those reference Lunar New Year.
Many people celebrate both! And why not? It's a heck of a good party.
One thing people do is make (or buy) mochi as an altar offering. Not everyone likes mochi, but it can be delicious. I can't get the correct rice, and I can't get sweet bean paste, nor even the beans to make it, but I am persistent and determined beyond all reason, so I made my altar set using Mochiko, or sweet rice flour. Most of the mochiko recipes I've seen are the Hawaiian variation that is very sweet, made as a confection. I prefer the version with just a touch of sugar, so that, when the celebrations are over it can be either diced, fried and seasoned with soy sauce and maybe some seaweed to make those tasty little rice crackers or, my personal favorite, sliced, grilled until it's nicely puffy and colored, and served with a dash of soy sauce. Here's my handmade altar set. It's small so there isn't room to put the mikan (Satsuma tangerine) on top, so I improvised.
I wish you all a wonderful year!