The reason libcairo2 is unavailable is because 7.10 doesn't include it in its package list. You could manually install this package from Ubuntu's package search tool, but there is probably a really good reason why libcairo2 isn't in your version's package list. And that really good reason is probably that you'd have to upgrade half of your system manually in order to install it. For example, (in the worst case) it might depend on a later version of glibc or libx11 than you have. You will almost certainly have to upgrade several non-critical packages that it relies on, as well.
Upgrading your whole system via the update-manager (in a controlled, automated way) is much better than upgrading a large portion in an ad-hoc, manual fashion.