Yes, the update will fit fine on internal flash. However, the NXE is split into two portions - there is the dash update side which fits onto internal flash and is only 8MB and then when you sign into Live, you'll be prompted to download an update. This is 120MB or so and is for the avatar side of things.
You can still boot into the NXE with no hard drive or memory card attached at all. You just won't get the avatar stuff and if you have a 64MB memory card with a Live-enabled profile on, it won't be able to sign into Live because it can't take the 120MB avatar update.
Again though, the basic dash will work absolutely fine with no storage devices attached. So, in order to answer your original question, you will indeed be able to download the update on one console then plug the HD into another console to apply the basic dash upgrade minus avatars.
