I've never tried doing it, but from my experience with getting flowers to stick with cakes, I'd say that part of the trick is the right consistency of icing, and working fast enough before the icing starts to set up.
I'd probably apply the icing in three stages -- a crumb coat, let it set up, then a medium thickness icing so the cake doesn't show through, let it set up, then a quick thin coat of icing for the glue for the sprinkles. This way, you've already got the cake coated in icing, and aren't wasting time trying to make sure you have proper coverage, and it's not soft and think, making it so the whole side is trying to drip down.
For the actual application of the sprinkles, I'd work over a sheet pan to catch the amount that's going to fall off, and then take a handful and just try to pack it onto the side until it doesn't seem to be taking anymore, regularly collecting up the fallen sprinkles to re-apply. Once I did the sides, I'd dump the rest of the sprinkles over the top, pack it on like the sides, and then shake (or brush?) off the loose ones that didn't stick.
update Just saw @hobodave's answer (why I didn't get the 'new answer' message, I have no clue) ... the parchment paper is a great idea for collecting the unstuck sprinkles to reapply ... I'd still set it in a sheet pan, to keep them from rolling off the side.