Because of the way that an Alcubierre drive alters space to get a ship to exceed $c$, unwarping that space (in order to stop/travel at sublight speeds) would undo the fancy compression/expansion of space that allows the ship to travel above $c$. In short, the act of destroying the "warp bubble" would bring the ship's velocity down below $c$. The ship would then travel at some high fraction of $c$ until it stopped, jumped back to warp, etc.
As to how long/hard it would take to stop, the quickest and easiest way would probably be to reverse the direction of the warp bubble. This would stop it rather quickly (no numbers, as I doubt anyone has ever calculated anything exact), probably within a few "bubble-lengths". The rapid "deceleration" would likely wreak havoc on the occupants of the craft though.