New Year's is a proper noun because it refers to a specific holiday- much like Christmas and Independence Day are proper nouns.
In most Western cultures, people often go to late-night parties on December 31st (New Year's Eve) and stay up until midnight passes and January 1st (New Year's Day) officially begins. (And many people do not have to go to work on January 1st even if it lands on a weekday.) This is the context for the holiday that often gets simply called New Year's - where it is the coming new year that is being celebrated on the last night of the old year.
So, the following would be very natural to hear:
Alice: "What are you doing for New Year's?"
Betty: "I'm going to a friend's house party for New Year's."
Alice's question basically asks what Betty plans to do for the New Year's (Eve) celebration, but the word "Eve" is omitted because it is simply assumed. If Betty had plans that don't fit the assumed pattern, she might have to be more specific:
Alice: "What are you doing for New Year's?"
Betty: "Nothing for New Year's Eve, but I'm having brunch with my parents on New Year's Day."
It is also common for people to make a New Year's resolution during this time period. This is a resolution/vow/pledge made for the new year on New Year's. Technically, calling it a "new year's resolution" would be perfectly logical as it is a resolution for the new year, but because it is the cultural norm to make/start these resolutions on New Year's Day the phrase is typically named after the start day as a "New Year's resolution".
NOTE: My above explanation comes from an American English background.
In the comments, Colin Fine mentioned that New Year (without the 's) is the common/expected phrasing for it in British English. (e.g. "What are you doing for New Year?")
The new year is an entirely different concept; it is just a simple construction like "the summer" or "next week" and can refer to the entire coming year or the first few weeks of the coming year. It will often be used in situations where "next year" could just as easily function, but people only tend to use it late in the year and particularly in December when referencing events coming in January or so.
Lastly, I find there can be a lot of miscommunication issues when people from Eastern cultures use New Year as the direct translation for their Eastern holidays when speaking to Western audiences.
Cheng: "The warehouse closes for New Year on Monday; so you'll have to expect some shipping delays."
Danny: "That's fine. It doesn't make much of a difference if it ships on Tuesday instead."
Cheng: "..."
The problem here is that Cheng is describing 新年 which is understood among Eastern cultures to be a multi-day (multi-week?) sort of "holiday season" celebrating the new year, rather than the sort of single-day "holiday" that Danny assumes as a Westerner when he hears "New Year". So, directly translating this concept as "New Year" creates an issue because the name too closely overlaps with the Western default. (In other words, a naive Westerner might think that "Chinese New Year" is exactly analogous to "[Western] New Year" just on a different, non Gregorian, calendar.)
One way to perhaps get around this issue would be for everyone to specifically start romanizing it as Xinnian in much the same way that Hanukkah and Ramadan are romanized. This would eliminate a lot of the half-translation issues (where the words translate but the cultural background does not) by simply encapsulating the concept into a new loan-word. Unfortunately, that would be language-specific to (Mandarin) Chinese and would mean that a single term likely wouldn't suffice for all the other similar Easter New Year festivals... resulting in the necessity of individual words for the Korean, Thai, Tibetan, Vietnamese, Japanese, etc. versions.
A different way to go would be to use a translated phrase common to all the Eastern traditions, but distinct enough from "New Year" to be properly differentiated. Phrases like "Lunar New Year" or "Spring Festival" might work better than simply "New Year" in this case, but the cultural context that these are multi-day events is again not given explicitly. Something like "Lunar New Year Festivals" would seem to me to be the most universal name... but unfortunately it is quite wordy.
Relating to that Eastern/Western divide is the specific example you brought up:
I often go to Japan in New Year (every the beginning of a year).
This phrasing would sound odd to most native English speakers; mostly because "in" is a strange preposition choice here - "for" or "on" are the more natural choices depending on what you're trying to convey. My guess is that you are trying to use "New Year" as directly-translated vocabulary to describe something of an "Oshōgatsu" tradition.
I often go to Japan for New Year's.
I often go to Japan for Oshōgatsu.
Either of these choices would be fine, but the meaning is slightly different. The first means that the purpose of your visit is to be in Japan to go to a Dec31/Jan1 "New Years" holiday party, whereas the second means that the purpose of your visit is to be in Japan for the purpose of being with family for Jan1/Jan2/Jan3/etc.
Of course, very few Westerners would know what Oshōgatsu is and you may find that you have to give a quick explanation like "the Japanese New Year holidays" or something similar every time you use the word... but saying "New Year" instead would miscommunicate what you are trying to convey.
Here's a sentence that combines most of the different versions as an example of how they're differently used:
I often book a flight to Japan on New Year's so that I can start the new year with my family for Oshōgatsu.
