How do I remove patches from my WoW client?

You answered your own question! :)

Yes, go into your patches and look for the most recent patch. Delete this and try, but make sure you haven't emptied your recycle bin, just in case something goes wrong and it turns out you've backed yourself into a corner. Private realms are a little tricky, but they should have been updated on their own. If not, this is a mistake of Blizzard's and not yours. Don't panic.