Imaging a person in an elevator accelarating towards earth with accelaration $a$. At some point the person let a ball fall from his hand. Does the time that ball needs to hit floor equal time that ball would need to hit floor if elevator wasn't moving?

No. Say for simplicity that the elevator is accelerating towards Earth with acceleration $g$. Assuming we're sufficiently close to Earth (but not too close that our experiment will end too abruptly!) anything inside the elevator is also falling with acceleration $g$ due to the Earth's gravity.

If you were to let a ball go in the middle of the elevator, it would begin accelerating downwards with acceleration $g$ due to the Earth's gravity. But the elevator too is falling with acceleration $g$, and so the ball and elevator are moving with the same speed/acceleration at all time (until we hit the ground).

What this looks like for someone inside the elevator watching is the ball simply "floating" in mid-air.

To answer your question

"Does the time that ball needs to hit floor equal time that ball would need to hit floor if elevator wasn't moving?"

the answer is "no", because the elevator floor is moving away from the ball, so it would have to take longer for the ball to reach it (and if the elevator and ball are falling at the same rate, the ball will never reach the floor).

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    Indeed if the elevator was accelerating with a downward acceleration which was greater than the acceleration of free fall $g$ an object which was released in the elevator would accurate upwards towards the ceiling of the elevator. – Farcher yesterday

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