It's also noteworthy that the distances we're talking about with the Oort Cloud are like...one light year or more from us. Most comets don't travel anywhere near lightspeed, so this should give us quite a while to prepare (like...we're talking a hundred to a thousand years for most of the ones that'd come our way).
Also, there's a star 4 light years away -right now-. Yeah, 1-2 light years away is considerably closer, but it's hardly a collision course.