Austria-Hungary can be formed by Austria by the decision The Austro-Hungarian Compromise. It will add Hungarian as accepted culture, but will remove the possibility to form Germany from Austria. Austria needs to have under 80 prestige and have researched Nationalism and Imperialism to make the decision. In addition, Italy and Germany both need to exist to form the dual monarchy. Austria-Hungary gains cores in all of Austria, while Austria loses all of its cores, except in those provinces where South German is the main culture.
Austria starts in 1836 with more than 80 prestige, which means that one will actually have to lose prestige in order to form Austria-Hungary. Although this sounds bad, it has large benefits like more national focuses and larger manpower. There are multiple ways of losing prestige. The simplest is to release puppets, which will give -5 prestige pr. nation released. Make sure not to release any nations with Hungarian cores. Venice, Trieste and Bohemia-Moravia are all suitable. Otherwise one could willingly lose wars.
The strategy of Austria-Hungary largely follows that of Austria, except that Austria-Hungary cannot form Germany. It will likely face the Austro-Prussian War.
Preventing the formation of Germany is a crucial, but difficult task. The easiest way is to outright annex a German minor e.g. Bavaria or Saxony (which however starts in Austria's sphere). Another way is to reduce Prussia to secondary power status by completely wiping out their army and occupying everything. It will lose its entire sphere, and the other Great powers will come in to sphere the German minors.