Use the fiber bundle $S^1 \rightarrow S^\infty / \mathbb{Z}_p \rightarrow \mathbb{CP}^\infty$ to compute $H^{*}(K(\mathbb{Z}_p,1);\mathbb{Z}_p)$

Question

Compute $H^{*}(K(\mathbb{Z}_p,1);\mathbb{Z}_p)$ using the fiber bundle $S^1 \rightarrow S^\infty / \mathbb{Z}_p \rightarrow \mathbb{CP}^\infty$.

Any takers? I have had a couple of ideas (e.g. Using the Hurewicz map), but nothing conclusive. I fear I may be lacking some information.