As a native speaker of Ukrainian, I would never use those two sentences like this, as a single extended cheerful way to express I'm happy because of the new year.
"З Новим Роком!" = Happy New Year! - you congratulate someone else.
"Новий рік!" = "Yeah! It's a New Year!"- you can say it to yourself, I guess. You can also use it with ура = hurrah! ("Ура! Новий Рік!")