Whether or not Thomas Mann was directly influenced by Jungian thought, it's not difficult to find the application of the concepts you name in your question to Death in Venice.

Aschenbach's personality is one in which there has obviously been a severe form of repression. We're led to understand that he has kept his emotions in check throughout his life and that his controlled behavior has been a counterpart to his art, in which the Apollonian (or "classical") element has been dominant rather than the Dionysian or emotional tendency. All of this control comes apart when Aschenbach visits Venice and becomes obsessed with the young boy Tadzio.

Jung's anima represents the feminine spirit that is unconscious in a man, as the animus is the counterpart masculine spirit within a woman. We can interpret Achenbach's interest in Tadzio as a manifestation of the anima, but in this case (if not overall) Jung's theories may appear to be just an intellectual construct to explain that some people are gay. Similarly, the shadow in Jungian thought is the negative unconscious, things repressed because they represent the undesirable aspect of the personality. Aschenbach at first (and understandably so, given the moral and societal strictures of his time, and simply the fact of the boy being underage) fails to recognize his feelings for Tadzio as what they are—a sexual attraction. When he does fully understand those feelings, his personality disintegrates. His death is ostensibly the result of the cholera epidemic that has taken hold in Venice, but we can just as readily attribute it to the emotional shock of Aschenbach's recognition that his whole life has in some sense been a lie, given that he's spent a lifetime denying and suppressing his true nature.

The prelude to Aschenbach's death is the terrifying dream he has of an orgy in which the civilized facade of European life has fallen away and people have become "savages" worshiping an alien god. The cholera epidemic, described as a plague of apocalyptic proportions sweeping across the Eurasian landmass from East to West, is a metaphor of this spiritual collapse, in both Aschenbach individually and in the society as a whole. Aschenbach's experience of this, revealed in the nightmare, is thus a manifestation of the collective unconscious. Mann's implication is arguably that European society is collectively aware that its civilized qualities are merely a facade, or at least that the potential for violence and savagery are always present and poised to burst forth. Death in Venice, like other works of literature and in the arts in general of the time, was an ominous warning that a cataclysmic series of events would occur—as they did very soon in World War I and not long after that in the Second World War and the Holocaust.