The premise is highly unlikely. Austria was well aware that she would never reapy Russian support in Hungary with support in the Balkans. Russians occupying the Danubian Principalities (as they did in the first part of the Crimean War, before Austria threatened war) were a big no go for Vienna.
I can imagine a situation where Austria allies with Russia, provided they can agree on a partition of European Turkey that was highly problematic for both; Austria actually was not so sure she WANTED a further share of Balkan Slavic peoples (notably Serbs) quite not welcoming her rule. They would upset an already problematic nationality balance (though it was not the clusterfuck of later times) and even give Russia some possible leverage to meddle inner Austria affairs. On the flip side, they could help weaken those obnoxious Hungarians, and their nationalism, albeit developing, was still infant.
However, Austria would gain troublesome mountanious and unruly places, a headache in Italy, and the risk of further headaches in Germany, for the pleasure to give Russia full control over the Straits and the mouths of the Danube, that is two vital bottlenecks for Austrian economy. This, in case of victory.
War is against the first world power of the time (Britain), another Great Power, France (possibly the second or third one), and two minor allies both in a very good military position to force a two-front was being helped by the major enemies. The only neutral great power has been humiliated by Austria some years before and is perfectly situated to open a third front in case it seeks for revenge. Defeat may seem a not so far-fetched possibilty.
Sort of not looking like a particularly wise move short term.
(Of course, Austrian leadership of the time is not famous for its incredibly wise moves in general, so who knows).