I've been thinking about depressed/apathetic protagonists as I've been writing a work with a protagonist who tends towards that way. The difficulty with a depressed/apathetic protagonist is that we generally want a protagonist who drives the film: generally speaking, it gets hard to identify with someone who's sad sackdom is unrelenting.
Sure, a character may start out slow and sad, but in your average story, something will snap that person out. (Though, please god, we hope it's not a manic pixie dream girl.) So how do we write stories about characters who remain depressed or apathetic until the climactic moment?
I'm not the first person to say this, but I think one possible solution here is to give the depressed character mini-goals. Or even negative goals. Take Leaving Las Vegas: the protagonist's goal is to drink himself to death during one wild vacation. But if that's all he wanted to do, he could do that anywhere. And if that's all he wanted to do, he could do it alone. But instead the alcoholic, suicidally-depressed character has other goals: to get to Las Vegas, to connect with some human being; then when things get too close, he wants to find some way to piss off the woman he's connected with. (It's been a while since I've seen the film, so I'm going off vague memories and the wiki page.)
Another solution is to give the character a strong supporting cast. Again in Leaving Las Vegas, the suicidal alcoholic connects with a prostitute who inverts his situation: she wants to live but has a dangerous job. I'm not even sure what her overall goal is in the movie--which is one reason we know she's not really the protagonist--but she has lots of scenic goals: getting out of bad situations, getting jobs, getting the alcoholic to stop drinking.
Here's one last final thought on how to make a depressed/apathetic character interesting: beware of giving too much of their depressive interiority. Depressed people don't walk around all the time thinking about how depressed they are; but any problem like depression can lead to repetitive, "stuck" thoughts. If a character's usual response is "who cares?" or "I'm not getting any enjoyment out of this," that can get pretty tedious to hear about if this is the only thought of theirs we see. Since this character may not be making big external gestures, it might be tempting to give more of their interior life--but their interior life is boring if it's so repetitive. You can be repetitive in certain art/high lit works (say, Carole Maso's stream-of-consciousness of a dying woman, Ava); but in other works, I think you want to save those interior glimpses of depression for when they're most effective.
I'm sure I'll have other ideas as I play with my depressed protagonist (what fresh hell can I make for him?), but here's my initial thoughts.