OK, here is my answer with almost a two-years' delay. In general, no letter (even in a reputedly 'phonetic' writing system) gives absolutely exact sound representation.
Even in two languages/dialects sharing a same alphabet (cf. e.g. English, French and Finnish Vs, or Brittish vs American intervocal Ts).
Even vocal systems of Bulgarian vs Russian do differ (and, yes, Bulgarian is easier to read and to understand as compared to Polish or Eastern Slavic languages, but that's my own experience and experiences do differ, too).
In general, after some lessons in Mongolian you might very soon observe that:
1) The Лл is pronounced more like Innuit Ll, Welsh and Spanish LL [λ] rather than as it is in Russian;
2) consonants are articulated harder than in Russian (like in Bulgarian phonetic system indeed; closer to some Arabic emphatic [d], [kh], [tt], etc.);
3) unlike in Russian and Hungarian, and like in Finnish, a combination of two same subsequent vowel letters means a long vowel of same quality;
4) a short vowel at the end of a word is treated as Ъ ъ in pre-revolutionary Imperial Russian (that is, remains unpronounced; that is, Mongolian байна sounds more like [bain] not like [baina]);
5) Р р sounds softer at the end of a word (well, in a closed syllable at least) and is closer to Swedish Rr;
6) unstressed O o in Mongolian is still always [o] while in Russian it is pronounced closer to [a].
This is, of cause, not like reading in Japanese knowing some Chinese, or reading Sanskrit/Hindi knowing some Devanagari script, but these nuances do matter. Hope this will help.