After starting the engine, the oil pressure does not rise to the green arc if the throttle remains at idle. If more throttle is applied then the pressure does come to the green arc and remains stabilizes there. According to the checklist, the oil pressure should reach the green arc (50 PSI) after 30 seconds from engine startup (or 60 seconds at most during cold weather).
I don’t know enough about the Cessna, but certainly on my aircraft the oil pressure comes up within a couple of seconds when the engine starts, and stays in the green arc irrespective of the throttle position.
On the ground, the oil pressure falling below the green arc would be a reason to immediately stop the engine to avoid engine damage.