Definitely the wrong tree. The straw wouldn't cut down on transmitted noise from your neighbor, but it would reduce the echo/reverberation in you space.
Even tightly baled straw is full of holes. If it couldn't hold back water, it sure aint gonna stop air molecules from vibrating.
Here goes the broken record: You need more decoupling, mass, sealing and absorptive insulation to kill the noise.
If you're renting, you could try a piece of MLV or MLV + foam backing, hanging it loosely against the wall, but that will only work so well. It won't compensate for an inadequately built party wall.