This is a pretty complex question as what you say depends on what you want to convey or ask.
If you're asking what someone is planning for the upcoming year, you'd use "in".
Are you doing anything special in the new year?
Note, in this case, you do need the article.
If you want to ask about someone's plans for New Year's Eve, you could use either "on" or "for", depending on where you are.
Are you doing anything special for New Year's [Eve]?
Are you doing anything special on New Year's [Eve]?
Note that in this case you will not use the article and, in casual talk, the "Eve" is usually implied, though it can be interpreted to mean "Day" as well, though usually context will tell you which is meant.
I believe that, in British English, you could also use "at" for the above sentence, though it may not work if you're talking about a single day.