If $\pi \colon X \longrightarrow Y$ is a cyclic cover of complex projective manifolds of dimension $n$ branched along an ample divisor $B \subset Y$, then by a variation of Lefschetz Hyperplane Theorem one can show that the pullback map $$\pi^* \colon H^i(Y, \, \mathbb{Z}) \longrightarrow H^i(X, \, \mathbb{Z})$$ is an isomorphism for $i \leq n-1$ and is injective for $i=n$. See Proposition 1.11 in 

<cite authors="Lanteri, Antonio; Struppa, Daniele C.">_Lanteri, Antonio; Struppa, Daniele C._, [**Topological properties of cyclic coverings branched along an ample divisor**](http://dx.doi.org/10.4153/CJM-1989-021-3), Can. J. Math. **41**, No.3, 462-479 (1989). [ZBL0699.14019](https://zbmath.org/?q=an:0699.14019).</cite>

Actually Lanteri and Struppa prove the corresponding result for the push-forward map in integral homology, for a direct proof in cohomology see Proposition 1.1 in 

<cite authors="Wisniewski, Jaroslaw A.">_Wisniewski, Jaroslaw A._, [**On topological properties of some coverings. An addendum to a paper of Lanteri and Struppa**](http://dx.doi.org/10.4153/CJM-1992-013-8), Can. J. Math. **44**, No.1, 206-214 (1992). [ZBL0766.14012](https://zbmath.org/?q=an:0766.14012).</cite>

Using Poincaré duality, from this it follows that all Betti numbers of $X$ and $Y$ are equal, except possibly the middle Betti number, for which we have 
$b_n(X) \geq b_n(Y)$.

On the other hand, in your case we can easily find the topological Euler number $\chi_{\mathrm top}(X)$ by additivity. In fact the ramification locus $R \subset X$ is isomorphic to the (smooth) branch locus $S \subset \mathbb{P}^n$ and the restriction $\pi \colon X - R \longrightarrow \mathbb{P}^n - S$ is an unramified double cover, so we obtain
$$\chi_{\mathrm top}(X) = 2(\chi_{\mathrm top}(\mathbb{P}^n) - \chi_{\mathrm top}(S))+ \chi_{\mathrm top}(R) = 2\chi_{\mathrm top}(\mathbb{P}^n) - \chi_{\mathrm top}(S)$$
and this allows us to recover $b_n(X)$ as well.