Let $X\rightarrow \mathbb P^n_{\mathbb C}$ be a double cover ramified over a smooth hypersurface $B$ of degre $2d$. In the case of hypersurfaces of $\mathbb P^n$ one can determine the integral cohomology using Lefschetz hyperlane section theorem and universal coefficients theorem.
Q. Are there some simple techniques allowing to compute $H^k(X,\mathbb Z)$ for any $k$?
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.
Expanding a bit Jason Starr's comment: let $R_X$ (or $R_B$) the jacobian ring of $X$ (resp. $B$). Recall that if $X=V(F) \subset w \mathbb{P}(a_0, \ldots, a_n)$, the jacobian ring is defined as $$ R_X := \mathbb{C}[x_0, \ldots, x_n]/(\partial_0(F), \ldots, \partial_n(F)),$$ with the $x_i$ suitably weighted.
From Griffiths result (and its extension to weighted projective hypersurfaces) the cohomology of $X$ can be recovered by looking at some homogeneous slices of $R_X$.
In your situation, if $B=V(f_{2d}) \subset \mathbb{P}^n$, then $X=V(y^2+f_{2d}) \subset w \mathbb{P}(1^{n+1}, d)$. In particular it is trivial to check that $R_X \cong R_B$ and you can find all the cohomology informations for $X$ by looking only at $B$. (note that in general $H^{dim \ X}(X) \neq H^{dim \ B}(B)$, since you will have to look to different homogeneous components of the Jacobian ring)
This is not an area of expertise for me, so forgive me if I didn't understand the question properly and hence this answer isn't on point. I think the original reference for this might be Lazarsfeld's thesis. In Springer Lecture Notes #1092, there is an article by him called "Some Applications of the Theory of Positive Vector Bundles". In section 3 of that paper he proves that for any finite map of degree d, $ f: X \to \mathbb{P}^n $ , the maps on homotopy $ f_{*} : \pi_i(X) \to \pi_i(\mathbb{P}^n) $ are bijective for $ i \leq n+1-d $ plus more. Nice paper that certainly seems relevant to your question.
