Austrian troops had already taken Budapest before the Russians entered the war.
Magyar Nationalists had managed to alienate all the other groups in the Hungarian Kingdom, most notably the Croatians. The Austrians had already smacked down the Italians (Radetzky) and first line troops from there could be made available.
According to the history book I have here (A history of Eastern Europe - Crisis and change, by Bideleux and Jeffries):
"In the end the Magyars chose to surrender to Russian rather than Austrian force majeure in August 1849. But the Austrians bore the brunt of the campaign, incurring about 50,000 war deaths to the Russians' 543 (not counting 11,028 cholera victims). The Magyars also incurred about 50,000 war deaths."
This would seem to suggest the Austrians were quite capable of finishing the job themselves. Looking at the smackdown Savoy received in the same period, the Austrian army was obviously capable enough to perform against regular forces. The mostly irregular Magyar forces would have been at a further disadvantage.
Also, the long term viability of the Hungarian state would have been quite low: the Magyars seemed to do almost everything to antagonize the Croatians, Romanians, Slovaks etc.