What is the computational complexity of the PageRank problem?
As pointed out on the wikipedia page you cite, PageRank is simply computing the eigenvector corresponding to the maximum eigenvalue of the (modified) adjacency matrix of the web graph.  Since this is simple linear algebra, it should definitely be in FP, if not much smaller classes.

Part of the issue with Google's implementation is that the web is so large that its adjacency matrix doesn't fit into the memory of any single (or even any 10,000) computers.  So although the complexity of PageRank is technically in FP, the scale at which Google is doing it doesn't really fit into the realm of traditional single-processor complexity.  To discuss the complexity of actually computing PageRank on a graph as large as the whole web, you'd at the very least have to consider some sort of distributed computation model.