Mongolian in China is already written on old Mongolian alphabet.
However you should note that the spellings are archaic and represent Classical Mongolian pronunciation better than modern Mongolian (Khalkha or otherwise). If you learn Mongolian written in Mongolian, you will have to learn spellings and pronunciation separately.
Meanwhile, Cyrillic spelling is a quite straightforward representation of pronunciation (in Khalkha dialect). There's some differences mainly in which vowels get fully pronounced and which are reduced but difference is much smaller than between classical spelling and modern pronunciation.
So, if you learn it written on old Mongolian alphabet, you will pretty much learn it on Cyrillic as well - provided you already know Cyrillic in first place.
What I don't know however is if the government approved spellings will just replicate old spellings or is someone working on reforming that & bringing it closer to modern pronunciation.