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 1.1 in [1]
The corresponding dual result for the push-forward map in integral homology is also true, see Proposition 1.11 in [2]
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.
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References
[1] J. A. Wisniewski: On topological properties of some coverings. An addendum to a paper of Lanteri and Struppa, Can. J. Math. 44, No.1, 206-214 (1992). ZBL0766.14012.
[2] A. Lanteri, D. C. Struppa: Topological properties of cyclic coverings branched along an ample divisor, Can. J. Math. 41, No.3, 462-479 (1989). ZBL0699.14019.