When a schizophrenic man is released from a futuristic mental institution, specializing in curing your inner “demons”, learns that he is being hunted by an assassin, he must rely on his fragmented memories to survive.

You've just identified the core of your story.

Forget backstory and setting in this instance I suggest revers engineering your concept from the goal.

Goal - Stop the man running the institute.

Now ask; why does the MC need to do this?
The answer should be as a result of a specific event that happened to the MC.

A few suggestions would be;

The man conducted an un ethical experiment on the MC he now needs revenge and wants to prevent this from happening to others.

The man killed the MC's family or friends he now needs revenge and wants to prevent this from happening to others.

The man illegally lobotomised the MC's family or friends he now needs revenge and wants to prevent this from happening to others.

There are many more options I'm sure so pick the best one and use that as the inciting incident.
Use that to start your logline then describe your character briefly and write in the goal.

What he must do is find "the man" running the institute with the help of this assassin/agent. Now how exactly do I word something like that especially since this man is one of the biggest plot points and mystery of the film.

What Nir Shelter says. To merely "survive" is not good enough. Every protagonist in every movie wants to survive. More than merely survive, the protagonist MUST need to do something, achieve some specific goal that will ensure his survival -- and defeat the antagonist.

What must the man do in this story?