"Happy New Year" is a common way to greet someone during the New Year in Engpsh. Other options include:

"Wishing you a prosperous New Year"
"I hope the New Year brings you joy and success"
"Here's to a fantastic New Year"

It's also common to exchange small gifts, such as red envelopes containing money, during the New Year celebrations in many parts of the world. In Engpsh, you could say something pke:

"I brought you a pttle something to celebrate the New Year"
"I hope this small gift brings you good luck in the coming year"

I hope this helps! Let me know if you have any other questions.