Can $\mathbb{C}P^2$ be the total space of a principal $U(1)$-bundle?
Since the universal bundle of $U(1)$ is $S^\infty \to \mathbb{C}P^\infty$, then we would just need to find a 3-manifold $M$ and a continuous map $f : M \to \mathbb{C}P^\infty$ such that the pullback bundle $f^* S^\infty = \mathbb{C}P^2$, or prove that such thing can't occur, but I don't know how to proceed further.
Suppose $\mathbb{CP}^2$ were the total space of principal $U(1)$-bundle over some space $B$, so that there is a fiber bundle $U(1) \to \mathbb{CP}^2 \to B$.
Note that the map $\mathbb{CP}^2 \to B$ is surjective, so $B$ is compact. If $B$ is a manifold (or even just a CW complex), it follows that the homology groups of $B$ are finitely generated, and all but finitely many of them vanish. Therefore $B$ has a well-defined Euler characteristic.
As the Euler characteristic is multiplicative for fiber bundles, we see that $\chi(\mathbb{CP}^2) = \chi(U(1))\chi(B)$, but this is impossible as $\chi(U(1)) = 0$ and $\chi(\mathbb{CP}^2) = 3 \neq 0$.