As a native speaker, I find that it is very common for people to wish each other a "Happy New Year" well into most of January, especially if it is the first time that they have seen each other since Decemeber 31st. I hear this in conversation all the time, and I see it, written exactly as you have written it, often as well.
I am not sure if you are also asking if it is correct to write
"Happy New Year" instead of "Happy new year".
If that is part of your question, then I would say that because it has become sort of a stylized traditional greeting, that your usage was correct. It is written that way all of the time, in the same way we say the following:
- Merry Christmas
- Happy Thanksgiving
- Happy Valentine's Day
- Happy St. Patrick's Day
- Happy Memorial Day