So what do you do when you teach your kids something that you know will upset them? You prepare intermediate lessons, so they can work their way towards the information you want to share. You share other stuff, bit by bit, until they become accustomed to the idea without knowing it.
You feed them ideas so they start breaking out of their mental box bit by bit.
I see this happening in the world, especially in places like this one. Not everyone will be following along, but that isn't required. Many people don't understand the involvements of political decisions (the involvement of economic interests, the implications and problems that the "obvious" solutions have and why those are not realized etc.). Many don't even care, or want to understand.
For most people, it will be something novel, something world-changing...for a month or two. When the media die down, it will be just something else that changed in the world. Humans are very adaptable.
I agree that there will probably be a few fires around the world. Then again, there still are people who still think the universe is but a few thousand years old. These things have their repercussions, but all in all, life goes on, and today has never been the same as yesterday. We're moving forward, no matter whether we like it or not.
Eventually, even under the most adverse circumstances, this will sort itself out generationally. If the aliens were announced today, then in twenty to thirty years the people who will be running the governments and companies will be the kids of today, who will have known the aliens for most of their teenage and adult years.