It's hard find a job in web design and development field in Mongolia?

I don't have first hand experience in that field here, but I get the sense that it's probably a mixed bag.  There is no doubt a fast growing market for web designers here, but also consider that the local size of this field and the overall economy is still pretty tiny compared to most other nations.  But I think the biggest hurdle might be language and the level of one's Mongolian language skills.  If you remember your Russian Cyrillic alphabet, that will certainly help for reading and writing.  But I suspect you will need a decent working proficiency in Mongolian for most such work, as Mongolian language is still by far the norm for local web sites (foreign language versions of local websites are still the exception rather than the rule).  If you're determined and good at languages then this should not be a barrier, but I imagine you may want to take up some teaching job or other kind of work in the interim while you develop your Mongolian language proficiency.

Also be aware that the salary is likely to be significantly lower than you might have been used to in Europe, as is the case for most kinds of job positions here (though entrepreneurship and running one's own business is often a good path to take in Mongolia, if you're inclined towards that).  And as strange as it may seem for a web designer, you also might be effectively limited to the Ulaanbaatar region for that kind of work (it's where most of the IT action seems to be, and many Mongolians still may not be as comfortable with telecommuting and remote workers/freelancers as Western companies can be).  If none of that bothers you, then by all means go give it a shot.  Just have an alternate plan (or two) for other kinds of employment, just in case.  Hope that helps.