After I get my MSA signed by a Notary, is this a legal document in the eyes of the Judge? Do I have to file with the Court right away? I live in New Jersey.

A notary is merely a witness, that does not make it any more legal. It will need to be ruled on my the judge.

A notary is merely a witness, that does not make it any more legal. It will need to be ruled on my the judge.
Actually a notary is not a witness but instead an official who can administer an oath. Notarization also is proof of sorts that the person who supposedly signed the document actually did so.
What's an MSA, by the way?

Actually a notary is not a witness but instead an official who can administer an oath. Notarization also is proof of sorts that the person who supposedly signed the document actually did so.
What's an MSA, by the way?

Actually a notary is both. But in the context of this question, a notary is used as a witness to certify that the document was signed by the person supposed to sign it. The notary is certifying that they have witnessed the signature and checked the ID of the signator and are satisfied that the signator is who they claim to be.

Notarization does not make the document legal. A judge still has to accept it. Notarization just means the judge does not have to witness the signing.