(Note: don't bother entering your AppleID during the installation process - it doesn't seem to accept it there. When the first menu appears, open Disk Utility and 'Erase' your virtual hard disk. VirtualBox 4 supports DMG files natively, and InstallESD.dmg is bootable. dmg to the file-picker dialog in VirtualBox). (I also suggest giving it more than 2 GB of RAM, and more than 1 CPU.) Turn the video memory up to 128 MB - at the default of 10 MB, it'll boot but you won't be able to see anything. You can use either 'Mac OS X (64 bit)' or the 10.8-specific one - they both seem to work. On OS X 10.9, you'll get a warning that you can't downgrade, which is fine. Even though they're just emailing you a PDF with a license code, it's done manually (so I'm told), so it really does take 1-3 business days to get to you. Go to Apple's web store ( not the App Store app), and search for 'Mountain Lion', and pay $20 for it.
First, you need to buy Mountain Lion (if you haven't bought it already).
