If a map $f:X\to Y$ of CW-complexes induces an isomorphism on homotopy groups it is a homotopy equivalence by the Whitehead theorem. In particular it also induces isomorphisms on homology and cohomology.
If now $f$ is just an $n$-equivalence (induces isomorphisms on homotopy up to degree $n$), intuitively I would say it still induces isomorphisms on (co)homology in lower degrees. But does it? Or is there a similar statement?
If $f$ is a $CW$-map, then $f$ induces an equivalence between $X[n]\rightarrow Y[n]$ where $X[n]$ is the $n$-truncature of $X$, you deduce that $H^*(X[n],.)\rightarrow H^*(Y[n],.)$ is an isomorphism by Whitehead. $H^*(X[n],.)=H^*(X,.)$ for $*<n$. To see this, use cellular (co)homolgy since all the maps above are natural, you deduce that the morphism $H^*(X,.)\rightarrow H^*(Y,.)$ is an isomorphism for $*<n$.