Learn English – the word for putting an image in someone’s head by describing it

You could say that they were resigned to their fate.

From Wiktionary:

To submit passively; to give up as hopeless or inevitable. [from 15th c.]
Here is a man who was resigned to his fate, who was walking to the scaffold and about to die like a coward, that's true, but at least he was about to die without resisting and without recrimination. Do you know what gave him that much strength? Do you know what consoled him? Do you know what resigned him to his fate?

Despite the question title, and the OP brother's phrase:

He cares but he doesn't care

There is nothing in the actual description which suggests the person does care, the OP describes himself as: I'm extremely passive as long as something doesn't directly affect me. which hints at apathy, rather than passivity.

2. lack of interest in or concern for things that others find moving or exciting. (RHD)

The noun form would be apathetic

A less derogatory term, would be detached (CD)

2. having or showing no bias or emotional involvement; disinterested

Overall, a sense of lacklustre, weariness and ennui (OD) is conveyed in the post.

A feeling of listlessness and dissatisfaction arising from a lack of occupation or excitement