Monkey Park Nagano New Year

Is the monkey park in Nagano open during new years time?
Between 1.-3. January

Yes, unless there are special circumstances. (For example, they were closed for a while in August because some of their staff had gotten corona virus.) If you are planning to go there, check their web site beforehand. Their home page is here: https://en.jigokudani-yaenkoen.co.jp/

Here is the park’s page that states they don’t have any “fixed holidays”:
https://en.jigokudani-yaenkoen.co.jp/guide

The Japanese version is even clearer (休苑日はありません。年間を通して無休です。)