The immediate consideration in buying an established business of any sort is you inherit their reputation.
This can be great, or a big challenge.
I've never bought a hostel, but the one I've managed for the last two years had some 'reputation problems'. Much of this stuff takes time, patience and persistence to change/fix. We're still not where I want to be with everything.
On the flipside, the first hostel I worked at had such a great reputation marketing wasn't necessary. Word of mouth did everything.
I'm happy to PM you specifics, or talk in more general terms here if you want more info.