Let us assume that a train of 1000 tonnes is moving with a constant velocity [so, [itex]\frac{dp}{dt} = 0[/itex] (p = momentum)] on a rough surface in the [itex]\hat{i}[/itex] direction.

Now, i apply a small force, of a very small magnitude, in the [itex]- \hat{i}[/itex] direction i.e. in the opposite direction of the motion. Will it cause the train's direction to reverse? Here, the force applied by the engine is just enough to counteract the force due to friction. So, even a small force [which i can myself] apply on the train, can i reverse the direction of the train?

I'm asking this because it seems quite like a paradox to me. The 1000 tonnes figure is more or less for the perceptual impact :D.

The rate of change of the momentum would be the the -i direction (for as long as the force is applied), but the momentum itself would still be in the +i direction, until the momentum reaches zero. Therefore you would either need to apply a very large force, or apply a small force for a very long time to actually reverse the direction of the train.

While you may be making the net force on the train point in the opposite direction as its motion, that doesn't mean it'll instantly reverse direction. To change the large (let's say) positive velocity to a negative velocity will take a large acceleration or a small one for a long time (as nicksauce said).