It all depends. As an author you can choose either way. A number of related questions:
New inventions and new phenomena will require new words. This is inevitable. But how much change is there in this society? That is up to the author. There is also possible to make old words have new meanings. ("to google" vs "to search the net")
Does the immortal themself want to appear to be one of the people. This will lead to them adopting the changes that occur naturally. If they want to appear above the people they will try to keep their language unchanged, and mostly succeed.
Is there TV or some other way for the immortal to speak to the people? And will they listen? This will contribute to keeping the popular language close to theirs.
Does people in general like and want to emulate the immortal? This will give stability. Or does people like the idea that the upper class doesn't understand them? This gives change, like cockney.
In general, how large is the differences between the classes? Large differences and strict separations lead to separate languages. The immortal will then only influence the upper class.
Religion can play a big part. If there is a strong church with the immortal either in charge or worshiped, or both, the clergy will speak like the boss does, unchanging throughout the ages. It is not a given that the populace will follow the clergy in this, but it could happen.
In short, make it up as you go along and nobody can tell you different. It is your world.