How would you say "the visa is new for me", e.g. the consulate just made me a new one?
The same, "Viza este nouă pentru mine" (the way you have phrased it), but it may be more natural to say "Viza mea este nouă" or "Am o viză nouă".
I would say “consulatul mi-a făcut o viza nouă.” But that wasn't Karalamak's question.
By the way, if you make a mistake in a comment you can edit it using the DL desktop application. I don't think it works with the mobile application.
In English "the visa is new to me means to whole system or concept is something i ma not familiar with, if i want to say i have a new one then it would be "My visa is new" or "the visa I have is a new one."