The short answer is, "not much." In my estimation, and in the view of Habsburg military historian Gunther Rothenberg, the threat of the Tsarist Army crossing into revolutionary Hungary in 1848 was a more important factor in the eventual putting down of the Hungarian revolt than any battles undertaken by the Tsarist forces.

Since, from the way you've worded your question I'm assuming you know a decent amount about the events of 1848, I'll skip most of the exposition and give you the meat of the answer. The revolt in the Habsburg Italian possessions had been put down by Windischgraetz, allowing the Italian theater armies to be deployed to the Hungarian plain. While victories were being won in Italy, however, the Habsburg armies had lost an engagement to the Hungarian rebels at Nagy-Sarlo. Habsburg forces only stopped a potential invasion of Austria after the staunch defense of Buda by General Hentzi, whose defection the Hungarian rebels had taken for granted.

With the availability of the Italian armies' transfer to Hungary, even the simple numerical superiority of the loyalist Habsburg forces available for deployment in Hungary made the potential for a Hungarian victory "unlikely." However, according to Rothenberg,

continued Magyar resistance was seen as damaging to the imperial prestige and threatened to cause a resurgence of revolutionary spritis in Bohemia and elsewhere. Therefore, on May 1, 1849, Francis Joseph officially asked Tsar Nicholas I to intervene...

The Russian Army had already been partially mobilized in March of 1848 by the Tsar (a famous enemy of any revolutionary sentiment) and they passed through the Carpathians into the Hungarian Plain in early June of 1849. At this point, the Hungarians were "hopelessly outnumbered." Feldmarschallleutnant Johann Baron Haynau (Habsburg Army) defeated the Hungarians at Temesvar on August 9. Soon after, the main Hungarian rebel body under General Gorgey found themselves surrounded by the Austrians and the Russians at the nearby town of Vilagos and on August 13, surrendered to the Russian Field Marshall Prince Ivan Paskievich.

In short, the Russian troops sent by Nicholas into Hungary in 1849 had a largely psychological impact, causing widespread fear among Hungarian rebels of the possibility of "pan-Slavist" uprisings among the Kingdom of Hungary's Slav minorities. Although the Russian forces did indeed help to encircle Hungarian troops at engagements throughout 1848 and 1849, they did not fight many large scale engagements, a fact for which they were later lambasted by their Austrian "allies."

Sources:

Rothenberg, Gunther. The Army of Francis Joseph. Purdue University Press: West Lafayette, Indiana (1976).