i've got two coats on and my taxi is booked, so here goes-
Assuming that it is possible to get the ball to travel straight up and down, ignoring the practical problems of the moon, air resistance etc. then the ball is being subject to a constant force causing an acceleration of about -9.81m/s^2.
As velocity is a vector quantity that is always changing to try to calculate 'time spent at velocity=0' is impossible to equate.
Actually, i suppose at a finite level, it is impossible to equate 'vel=anything' at any instant in time. This could only really be evaluated during 'delta t' however you choose to define this quantity.
At best, it is only possible to approximate the standstill time in relation to human perception, i.e. <1 of those planck thingies.
Again, due to human perception and the finest possible resolution we can theorise over, as the smallest amount of time prior to the elusive Vel=0 and the smallest amount of time following vel=0, we could say that to adopt some kind of tangible answer, the apparent standstill time would be between 2 to 3 planck thingies.
In reality it never stops
i suppose its the same as trying to calculate the pressure experienced by a surface when a given load is present via a point contact.
Does that depend on how sharp the point is, or how small you can define a unit of length? What would happen if you could actually manufacture a true point?
off to the pub now