What is the change of momentum (impulse) applied to a baseball when thrown from rest with a 250 N force (5 kg mass accelerating at 50 km/hr)?

First, if a 5 Kg ball accelerates to 50 Km/Hr that does not mean that it was subjected to 250 N. Remember that F=ma, not mv.

The impulse applied to the ball is defined as , which equals change in momentum . You have values for m and delta v, so you can calculate the impulse directly.