Q:

Congratulate someone on the forthcoming new year

In Russian we say, "С наступающим Новым Годом!" What's the English equivalent? How do you congratulate a person before New Year comes?
After 23:59:99 on the 1st of January we can say Happy New Year - С Новым Годом!

A:

It's just plain ole 
Happy New Year!
Anglophones begin wishing each other Happy New Year more or less the same time as Christmas.

Merry Christmas and a Happy New Year!

Google images
Hmm... I can sense the OP's disappointment
OK. In Scotland the last day of the old year is called 
Hogmanay

The roots of Hogmanay perhaps reach back to the celebration of the winter solstice among the Norse, as well as incorporating customs from the Gaelic celebration of Samhain. The Vikings celebrated Yule, which later contributed to the Twelve Days of Christmas, or the "Daft Days" as they were sometimes called in Scotland. Christmas was not celebrated as a festival and Hogmanay was the more traditional celebration in Scotland. This may have been a result of the Protestant Reformation after which Christmas was seen as "too Papist"

Wikipedia

