How do I stay idealistic in academia without becoming cynical?

You are trying to guard the real intellectual work while the politics and pettiness of the institution keep seeping into how you see everything. That creeping cynicism is not a character flaw. It is what happens to people who came for something genuine and keep bumping into something smaller. The disappointment is proof you still care. But cynicism is a trap dressed as wisdom. It feels like seeing clearly, when really it just lowers your eyes so nothing can let you down. The institution and the work are not the same thing. You can lose faith in the first without surrendering the second, and keeping them separate is the task. When the cynicism creeps in, point yourself back at one concrete piece of real work: a question that still fascinates you, a student who lit up. Do that thing today. The institution will keep being the institution. Your reason for being here does not have to shrink to fit it.