You have recently graduated and are now applying for a Schengen visa in order to take courses in Hungary.
I'm gonna apply for a Schengen visa but what should I mention as my
occupation since I graduated and I can't have a proof of ongoing
You can put down "Prospective Student"
and does this affect the visa process?
Everything you put down on the application affects the outcome, and 'Occupation' is no different. Your 'occupation' tells them what to look for in assessing your personal circumstances, so don't over-decorate your occupation beyond what it actually is or they will worry if you are presenting yourself accurately.
"Prospective Student" is fine.