The main political problems that the Austrian Empire faced--emerging liberalism and nationalism--went largely hand in hand, and Metternich definitely understood them that way. As far as his success in dealing with them, his main contribution was to encourage collective action on the part of other European monarchies to crush uprisings motivated by liberalism, as he and Tsar Alexander I did in response to uprisings in Spain and the Kingdom of the Two Sicilies in 1820-1821. This was the system known as the Concert of Europe, and was first agreed upon in principle at the Congress of Vienna, where he was really the most active diplomat. Most famously, he persuaded the German Confederation to crush dissent in the universities and press through the Carlsbad Decrees.
It was nationalism that was Metternich's biggest problem in the Austrian Empire, which was composed of dozens of different ethnic groups who wanted some degree of autonomy from Austria. When revolutionaries in Hungary demanded reforms, including self-rule, young revolutionaries took to the streets of Vienna, and Metternich fled the country. But eventually the Austrian government crushed the revolutionaries, and the emperor called on Nicholas I of Russia, whose predecessor Alexander had been a major ally of Metternich's, to send in troops to end the revolution in Hungary.
So, Metternich's legacy was mixed. On the one hand, he created the system, institutionalized by the treaties of the Congress of Vienna, that withstood even the revolutions of 1848, at least in Austria and Germany. On the other, he was unable to permanently hold Austria together, and many of his policies, most notably supporting the Ottoman Empire against the Greeks and later the Egyptians, weren't tenable in the long run. He saw himself as a failure, but his system did a lot to avert a continent-wide war during his lifetime, which ended just as Bismarck was coming on the scene to take his place as Europe's foreign policy "mover and shaker." But ultimately, nationalism and liberalism, two forces he really did not comprehend, were bound to win out.
Any success Metternich had was short lived at best.Metternich was particularly determined to preserve the old Order. Working through the German Confederation, Metternich had the Carlsbad Decrees issued in 1819. The decrees required the German member states to suppress subversive ideas in universities and newspapers. A committee was established which employed spies and informers to investigate and punish liberal or radical organizations.
Born into a noble family, Metternich believed in his heart that the most stable and proper arrangement of society was a judicious combination of monarchy, aristocracy, and respectful commoners. He also believed that liberalism, as typified in the American and French Revolutions were responsible for the period of warfare, bloodshed and suffering that had just ended. Those who stirred up liberal ideas such as representative government, he believed, were engaged in a conspiracy to destroy the existing order, and were inclined to stir up the lower classes who in their hearts really wanted to be left alone. Liberalism, which Metternich opposed, was closely aligned with Nationalism. Liberals believed that each national group had a right to establish its own independent government and seek to fulfill its own destiny. Metternich bitterly opposed this idea, as it not only threatened the existence of the aristocracy, but also threatened to destroy the Austrian Empire and revolutionize central Europe.
It is important to understand the complexities of the Austrian Empire to appreciate Metternich’s position. The empire was a polyglot of several nationalities. It was dominated by German speaking people, but they were only one fourth the population. Magyars dominated Hungary, but were also a distinct minority. In Bohemia and Moravia, Czechs were the major group. In addition, there were Italians, Poles, Croats, Romanians Serbs, Ruthenians, and Slovenes The latter groups together formed a majority of Austria’s population, but each group alone was only a small percentage. Linguistic differences were present in different provinces; with different languages often extant within the same village. This made Austria strong in a sense, because of its large population and vast territory; but at the same time made it weak because of its many nationalities, who all could potentially become dissatisfied with their status within the empire. To give rise to liberal sentiments, in which each of these groups might express nationalist sentiment or demand some other form of recognition was potentially disastrous to the Austrian Empire. Metternich had no choice but to oppose it.
Rebellion in the Austrian Empire ultimately begin in Hungary, where nationalist Hungarians demanded national autonomy, full civil liberties, and universal suffrage. When the monarchy hesitated, students and workers took to the streets, and peasant revolts broke out in various parts of the Empire. Faced with widespread rebellion, the Emperor, Ferdinand I, promised reforms and to grant a liberal constitution. Metternich, the unrepentant conservative, saw the handwriting on the wall and fled to London in disguise.
Different people can answer this in different ways. It depends on whether you think that Metternich did well just to keep Austria in power from 1815 to 1848 or whether you think he failed because of the revolutions of 1848.
In 1815, Metternich seemed to do a good job of protecting Austria's empire from the wave of nationalism that was sweeping over Europe. He was able to get Austria a dominant position in the German Confederation and he was able to help prevent the rise of a united Germany at that point. By doing this, he helped suppress nationalist feelings that would have split the multi-ethnic Austrian empire.
Of course, Metternich's efforts could not prevent the revolutions of 1848. Rebellion broke out around the empire and Metternich was forced to resign and flee. This, clearly, means that he was unable to deal with the problem of nationalism within the Austrian Empire at that point.
Overall, then, Metternich dealt with nationalism (the main threat to Austria) well enough to allow for 30 years of relative stability.