Well, let's look at it another way. Imagine a t-ball is resting and waiting to be hit from the tee. You swing the bat and there is a one-way* energy transfer (the ball has 0 momentum at rest) that gives the ball momentum.
Now we throw a ball at a brick wall. When it hits, the ball loses momentum* and falls to the ground.
Now, let's combine the two. The ball is at 0 k/h only before the bat makes contact, but only after it hits the wall. Is there any time between these two events? In the fly + train problem, there isn't. They're the same event. The only conclusion, then, is that if there is ever a time when the ball/fly is at 0 k/h, it must be at the exact moment of impact.
*Per Newton's Third Law of Motion, this isn't true, but we're not really concerned with the extra details.