I was corresponding with the director of Hustai Park and I asked him your question, and this is the link to the web site he suggested (all in one line):
Some details can be found also in the web site of the park:
If you will be there around the end of July, it would be nice to meet you there...
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I am still at the stage of collecting information and planning my trip (it will be a trip mostly with my students who study the wild ass in Israel), so I can't give you any tips yet on places etc. I surely recomend to people who plan to ride for a few good solid days in Mongolia to train well before that. One doesn't need to ride a similar 5 days ride (probably the wrong thing to do, if one is not trained), but building up the right muscles with weekly outdoors rides, gradually increasing in length and intensity will be a good preparation.
Talk to you later, when our plans are more down to earth and we have more info - lets communicate through private messages.