This probably isn't the most efficient way of doing this, but I guess you can get the probability of each of the orderings and work with that (i.e. and so forth, which I assume you know how to get). Probably that would get out of control for large n though. At the very least you can streamline those sort of calculations - you would be working with a bunch of multivariate normals - but then you would have to do of these.

I don't see any reason to think this is an easy problem in practice if you really want to get exact answers. You could Monte Carlo this if it is a practical problem.

At least this takes care of the example you gave though. You only need to calculate 6 probabilities to know the distributions of the rankings.