No, Zakaria does not at all think that the challenges faced by the US are insurmountable. He is actually fairly optimistic about the US's chances. The point of the book, in fact, is to argue that a "post-American world" is not one that has to be bad for the US.
What Zakaria thinks is that the US simply needs to be more flexible in a number of ways so that it can benefit from the "rise of the rest." He thinks the US needs a more flexible attitude towards the world, one in which we do not view the rise of other countries as a threat. He thinks we need to be flexible in how we see our role in the world. We need to think of ourselves as a facilitator for the rest of the world. We need to use our legitimacy to help the rest of the world solve their disputes instead of trying to use military force to impose our will.
He also thinks we need to be more flexible internally. We need to accept more immigration. We need to be more flexible in dealing with our political issues like the government deficit. We need to be flexible and willing to allow our government to help promote technological innovation so we do not fall behind countries that do such things.
All in all, Zakaria is not pessimistic about the future. He does not in any way think the US's challenges are insurmountable.