Answer: 160

Small Hint:

Substitute $u=z^3$

Big Hint:

The two possible arguments of $z^3$ are $120^\circ$ and $240^\circ$

Solution:

Let $u=z^3.$ Then $u^2+u+1=0,$ so $u$ has argument $120^\circ$ or $240^\circ.$ Taking cube roots gives possible arguments $$40^\circ,160^\circ,280^\circ,80^\circ,200^\circ,320^\circ.$$ The only one strictly between $90^\circ$ and $180^\circ$ is $160^\circ.$