Will an Alcubierre Drive arrive before a beam of light, if sent between two stationary observers? First, I'm not a physicist, just an engineer, and my first language is not English, so forgive any strange wording.
I have a simple thought experiment about Alcubierre Drives: Imagine two stationary observers A and B, located several lightyears apart. They have the ability to create Alcubierre Drives. Let's say A wants to send some information to B. A puts the information on a document in Alcubierre drive and sends it to B. On the same time, just as a reference, A also sends the same info via a good old radio signal.
What will B see first? The arrival of the Alcubierre drive or the arrival of the radio signal? If it's the drive, I guess it would mean that A has sent info to B faster than light.
Or an even simpler example: A and B want to communicate faster than light. So they send data bits (0 and 1) to each other, encoded by sending either one or two Alcubierre Drives.
Or have I completely misunderstood the idea behind these things? Is the idea perhaps that someone inside the drive will only age e.g. 1 year, even though the distance is 1000 lightyears?, and that the drive itself will still take 1000 years to arrive, from an outside stationary perspective?
 A: Firstly, I would like to emphasize that Alcubierre drive or similar warp drive concepts are highly speculative. They all work with negative enegy density which is possible (see e.g. Casimir effect) but the only in very small amount. In contrast warp drive needs huge amount of negative energy to bend space-time as required (the amount is similar to a whole planet converted to energy).
But let us assume that the warp drive works. It should allow to faster than light movement, however, there is no violation of the theory of relativity because it is space-time itself which is moving, not a ship (there is no restriction on "speed of space-time", however no information is transmited by this mechanism!). The ship is confined to the warp bubble which is moving. Effectively, the ship is moving faster than light and of course everything on its board, i.e. the documments or hard drive with data. The light beam is slower than the ship and as a result information in the documents on the ship board arrives to the target destination earlier than the light beam. This effectivelly leads to faster than light communication.
However, please keep in mind that there are many issues connected with the warp drive I mentioned above. What the Mr. Alcubierre did is that he imagined how the space-time should be bend to allow faster than light movement and then worked out whole theory. But this led to need of negative energy density (huge amount of the negative energy!) which is currently considered "unphysical".
EDIT (on causality):
As in any other case when we travel faster than light, we face causality violation. In case of FTL travel different observers do not agree on an order of events. The causality is not violated inside the warp bubble as there nothing exceed speed of light. However, imagine you have more ships, each moving FTL. Than observers on these ships do not have agree on events order, thus the causality can be violated outside the warp bubbles. Effectivelly, this can allow you to travel in time.
However, there is a chronology protection conjecture (conjecture, not the verified theory) by S. Hawking. In short, it says there is always some mechanism preventing you from traveling in time. The conjecture seems reasonable as time traveling would lead to many paradoxes. Concerning, the "preventing" mechanism, in case of Alcubirre drive, it could be the negative energy requirement.
If you want to learn more about the time travelling, warp drive and worm holes and similar stuff, I would recommend book Time Travel and Warp Drives: A Scientific Guide to Shortcuts through Time and Space. The book firstly discuss special and general relativity and then show if warp-drives and time traveling could exist from current state-of-the-art scientific point of view. It is writen in more or less plain language although some equations are not avoided.
