It might be easier to try the problem for smaller numbers of bottles first.
Source: Internet newsgroup rec.puzzles, Used as Ken's POTD 5/16/94.
Consider the bottle #1 . Over all possible permutations (N=9!), there is a one-in-nine chance that this bottle gets its correct label. Same result concerning each of the eight bottles #2, 3, 4,...9. In total, there are 9 * N/9 = N bottles correctly labeled, that means (N-correctly-labeled-bottles/(N-possible-configurations) = 1 bottle is labelled correctly on average.
The correct labelings can be viewed separately, since we are considering the total number correct labels over all possible permutations.