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A few years ago I saw an anime movie on Adult Swim.

This movie began with a guy walking into an empty city with a photo of a girl with an arrow pointing towards her boobs. Later in the movie he gets enrolled in this mech program to save the city from giant blue crystals that bleed red and were trying to destroy the city.

He later bonds with a girl who had a terrible back story and saves the town.

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This might not be it, but your description reminds me of the seminal anime Neon Genesis Evangelion. Or more likely, Rebuild of Evangelion, a series of movies (three so far, with a fourth in development hell) that retell the original story with a few changes.

This movie began with a guy walking into an empty city with a photo of a girl with an arrow pointing towards her boobs

The protagonist, Shinji Ikari, spends the series living with a woman named Misato who has a not-insignificant bust. She sends him a postcard so he'll recognise her.

young woman on a postcard, wearing a bikini

Later in the movie he gets enrolled in this mech program

Shinji does indeed "get in the fricking robot", as the saying goes.

to save the city from giant blue crystals the bleed red and were trying to destroy the city

The antagonists of Neon Genesis Evangelion are the Angels, bizarre alien entities that are trying to destroy the planet. One of them, Ramiel, is a giant blue screaming octahedron.

*SCREAMS GEOMETRICALLY*

He later bonds with a girl who had a terrible back story

That could be literally any of the girls in Neon Genesis Evangelion. Pretty much all of them have terrible backstories.

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