This is going to be a bit strange if you haven't seen it before. I'll give you some guidance through the solution and have you fill in the steps as you go.
Initially, t = 0, v(0) = const. (That's the best we can do for an initial speed without more information.)
I am going to neglect the weight of the object (so no mg in the problem) and I'm going to assume that the force on the sphere is going to be strictly in the horizontal direction as I am also going to assume that the drag force is horizontal. You can work without these assumptions, but the problem gets really nasty really quickly.
So. We know that
This is a bit bulky so I'm going to define
and our equation becomes
This is a separable differential equation. I'll leave the details here to you.
So doing the integration and a little cleaning up we have
(Recall the definition of before you quote your answer.)
For the last part let us make a new definition:
Then the rest falls into place:
x = dv/dt, integrate, solve this for t, then plug this into the equation above for t and there you have it.
Let's see how far you can go with this. Come back if you need more.