I have assumed this question is refering to the short story, "A Problem." I have edited your question to indicate this. Let us just remind ourselves that indirect characterisation is when we are given some clues about a character and their traits but we are told this information indirectly, through things they do or say or what others say about them. Note Sasha's response as he hears his family debate his future so vigorously:
Sasha Uskov sat at the door and listened. He felt neither terror, shame, nor depression, but only weariness and inward emptiness.
Note the way that this quote identifies Sasha's feelings of detachment and his apathy. He doesn't care about his future and has no emotional involvement in what is going to happen to him.
Note what he says in response to what he hears:
"If Siberia, then let it be Siberia, damn it all!"
Again, his lack of investment in what happens to him and where he might be sent reinforces his general sense of pessimism and how superemely indifferent he is abotu his future.
However, it is clear that he is not that detached, for he still cares enough about his name to desire to defend himself from false charges:
Sasha was indifferent, and was only disturbed by one circumstance; on the other side of the door they were calling him a scoundrel and a criminal. Every minute he was on the point of jumping up, bursting into the study and shouting in answer to the detestable metallic voice of the Colonel: "You are lying!"
This clearly does indicate that, as apathetic and detached as Sasha feels, his name and reputation is still important to him, as it provokes anger in him when he is slandered and a desire to defend himself.
Sasha is indifferent about his future, yet he still cares about his name being charged with false charges.