I am a turkish student. I do my placement now in Germany. My Schengen visa will expire in a few weeks. Can I prolong my Schengen visa or get a new visa for study in Hungary directly in Germany, or I have to go home to arrange it?

You can get a new Schengen visa in Germany. About the details you have to ask the Hungarian Embassy. It is the same situation in the other Schengen countries.