A big issue with your question, right off the bat, is that the answer is heavily dependent on the time you're talking about. You don't mention Hungary in your question (although it's there in the heading), so let's assume you're talking about Cisleithania (i.e., the Austrian part of the empire and not Hungary) and that you're asking about the period beginning in 1867, when the Dual Monarchy was established.

The short answer to your question then becomes "No, the Emperor (not the Archduke -- all royal princes were archdukes) did not wield absolute power -- in practice." The constitution for Cisleithania promulgated in 1867 afforded substantial power to the Reichsrat (imperial council -- equivalent to a parliament). While suffrage was fairly limited in 1867, by 1907, when the last large electoral reform was implemented, Austria had universal male suffrage -- before even the U.K. had it -- so it was largely a democratic state. That said, the Emperor reserved several prerogatives to himself, including leading the army, and he could dissolve the Reichsrat at will, although he didn't do so except in extraordinary circumstances. (The term "he" really applies in this case, by the way, since for all but two of the years we're discussing, the Emperor was Franz Joseph.) He could in theory do almost whatever he wanted, at least constitutionally speaking, but he was smart enough by 1867 not to try, given the territory that attempting to wield absolute power had cost him (Italian territory, as well as a more-or-less indendent Hungary).

Austria also allowed substantial local power to be wielded. The provinces, or crownlands, of Cisleithania elected province-wide legislatures (called Landräte) in which more local government was conducted. They were not independent, however, nor in most cases were the people running them royal or noble. And, more importantly, they were subordinate to the central government and therefore to the Emperor and his government, and there were limits to what they could do. A good example is the attempt at language reform in Bohemia in the 1890s. Czechs wanted the Czech language given equal status to German in that province, but doing so required intervention and approval from the central government. When Minister President (equivalent to Prime Minister) Badeni approved the measure, support by Germans for the government was withdrawn, and it collapsed.

Nationality was a consistent source of tension in the last five decades of the empire's existence, and the central government, and thus the Emperor, went out of its way to accommodate national aspirations while maintaining the integrity of the empire's territory. By World War I, the empire had nine recognized national languages, even if they could not all be used at the administrative level in the provinces where they were spoken. German speakers, or at least many of them, certainly wanted a special role and higher status in the empire, but by 1867, they had already conceded equality on that level with the Hungarians. Czechs and Poles followed rapidly thereafter in asserting their own national aspirations. Political parties were founded among German speakers to assert their dominance in imperial society, and some of these parties (particularly the Christian Social Party of Karl Lueger) were successful among German-speaking voters. But ethnicity mattered in politics and parties were by and large defined on an ethnic basis, so much so that, in the 1900 Reichsrat election, the single biggest party was the liberal Czech nationalist party. So only when they were sufficiently unified in political purpose could German speakers actually wield political power, and even then, they were hemmed in by both other national communities and by the Emperor himself.

Jonathan Kwan's book Liberalism and the Habsburg Monarchy, 1861-1895 is a good overview of most of this period. Although its treatment is focused on the Christian Social Party, John Boyer's Culture and Political Crisis in Vienna: Christian Socialism in Power, 1897-1918 provides a very thorough look at the remaining years.