Anyone can use it if they know the Voice, like the Greybeards and the ancient Nordic heroes who created it. The exception is dragons themselves; the concepts it embodies are so alien to dragons that they can't project them. Also, the Last Dragonborn isn't the only Dragonborn.
And I thought of something else: wish. The D&D spell, that is. It lets you literally rewrite reality. There's a creature - the tarrasque - that can regenerate itself from being disintegrated into its constituent atoms. Wish is the only thing that can put it down, and only then if it's as close to dead as it can otherwise get. Its resilience sounds a bit like what that elixir you're talking about grants.
If even that doesn't suit you (because of the different rules governing various continua), there's this other nice little thing called soul bind. You said that the elixir prevents its drinker's soul/body/whatever from being confined to the underworld/afterlife/whatever... but nothing about it not being able to be confined in a gemstone. It's basically a magical alternative to sticking her in a black hole - there is no escape.

Black holes, realistic ones at least, probably would work. Even if they wouldn't kill the Sue, she wouldn't be bothering anyone again. This happens for several reasons:

First, the Event Horizon. Absolutely nothing escapes once it's past the Horizon, not even light (that's why they're called black holes). The Sue sure wouldn't be able to.

Second, any realistic black hole has insane time dilation. Even if, by some miracle, the Sue did manage to escape, it would be in the far future (how far depends on how massive the black hole is).

Third, there's this Fun thing called Spaghettification (I'm not kidding, that's its actual name) that happens when you get too close to black holes. Spaghettification is when tidal gravity (the difference in gravity between your top and bottom) rip you apart. And then it happens again, and again, and again, until you're a cloud of disconnected atoms. Meanwhile, you're being squeezed into a tube of matter resembling a spaghetti noodle (hence the name). Even if the Sue had drunk the Hourai Elixir, this would happen to her. Heck, because she wouldn't be able to stop falling into the black hole, it would happen to her continuously.

One of the few ways this wouldn't work would be if it was banned as torture. Black holes are a very, very thorough way of killing stuff.

Find a way that works and would stay working, something that would freeze the Sue somehow but not kill her. That way she's stuck and can't reform or whatever it is.

Just need to make sure it's something that won't break...and then find a safe place to leave her.

~DF