It's possible visit the four peaks of Mt. Huashan whithout go by cable car?

You can go up the mountain without taking the cable car.
You need 5-6 hours to get to the mountain and it takes longer (perhaps another 2-3 hours) if you wanna experience all peaks.

Yes, your plan is good, but actually, you don’t need to stay one night in the mountain, it will be cold and there is no tent available but heavy long coat for rental.