In our time line, the secret to the success of National Socialism was its appeal to draw support from people who had previously voted for either right-wing ("nationalist") and left-wing ("socialist") parties. Had Rupprecht von Bayern become chancellor in 1933, he would have had an opportunity to deprive the National Socialists of a good number of their supporters.
At first, the National Socialists would probably respond to this by courting sympathetic senior veterans of the World War. However, as most of these were connected to either Rupprecht or Hindenburg (or both), I suspect that the best that the National Socialists could have done is to have their photos taken with Ludendorff. (Already several rounds short of a full magazine, Ludendorff was not the sort of fellow that the National Socialists wanted to put on a speakers' platform.)
The National Socialists might have done better with some of the younger "celebrity veterans." (They already had Hermann Goering.) However, I suspect that, when faced with a choice between Hitler and Rupprecht, most of these younger men would have chosen the latter. The same is true of those veterans who had published popular books. Hans Zöberlein, who had been an early convert to National Socialism, might have remained in the Hitler camp. Ernst Jünger and Walter Bloem, however, would probably have come out in favor of Rupprecht.
Having lost both the Veteranenkampf and much of their own right wing, the National Socialists would move to the left. Internally, the left-wing of the party, represented by people such as Ernst Röhm and the Strasser brothers, would gain power. Externally, the National Socialist German Workers Party might try to cooperate with other parties on the German left, particularly the Communists and the Social Democrats.
In government, Rupprecht is likely to have pursued the sort of authoritarian, corporatist, socially-conservative policies pursued by such contemporaries as Pilsudski, Smetona, and Dollfuss. These would have included policies aimed at preserving small businesses from competition from larger enterprises (such as department stores), support for family farms, and protective tariffs. In other words, Rupprecht would have sacrificed the possibility of economic growth on the altar of social stability.
In the realm of foreign policy, Rupprecht would have pursued a policy aimed at the gradual unraveling of the restrictions of the Treaty of Versailles. Making use of his many international contacts, Rupprecht would have been able to reassure the victors of 1918 that a strong Germany need not be a threat to the peace of Europe. Indeed, in a world in which the Soviet Union was devoting enormous resources to building up its military power, Rupprecht would have been able to paint a picture of a Europe in which Germany served as one of the pillars of an anti-Soviet alliance.