Here is another interpretation of the wrong way map in cohomology.

Let $f:X\to Y$ be a proper continuous map of "nice" topological spaces (say, of the form "a finite CW complex minus a subcomplex"). First, recall that the usual pullback map $H^\ast(Y,\mathbb{Z})\to H^\ast (X,\mathbb{Z})$ can be constructed as follows: we identify $H^i(Y,\mathbb{Z})$ with $Hom(\underline{\mathbb{Z}}_Y,\underline{\mathbb{Z}}_Y[i])$. Here $\underline{\mathbb{Z}}_Y$ is the constant sheaf on $Y$ and $Hom$ is taken in the bounded derived category. Then we apply the pullback functor $f^{-1}$ and use the fact that the pullback of the constant sheaf is constant.

Something similar can be done if we are looking for a map in the opposite direction. Namely, let us identify $H^i(X,\mathbb{Z})$ with $Hom(\underline{\mathbb{Z}}_X,\underline{\mathbb{Z}}_X[i])$.
Applying the (derived) direct image functor we get a map from $H^i(X,\mathbb{Z})$ to $Hom(f_{\ast}\underline{\mathbb{Z}}_X,f_{\ast}\underline{\mathbb{Z}}_X[i])$.

Moreover, there is a canonical map from $\underline{\mathbb{Z}}_Y$ to $f_\ast f^{-1}\underline{\mathbb{Z}}_Y=f_\ast \underline{\mathbb{Z}}_X$. So we get a map from $H^i(X,\mathbb{Z})$ to $H^i(Y,f_\ast\underline{\mathbb{Z}}_X)$ and the problem is to construct a map from $f_\ast\underline{\mathbb{Z}}_X$ to $\underline{\mathbb{Z}}_Y$ or some shift of it.

In some particular cases such maps indeed exist. For instance, if $X$ and $Y$ are smooth orientable manifolds of dimensions $n$ and $m$, then we can take the map $\underline{\mathbb{Z}}_Y\to f_\ast \underline{\mathbb{Z}}_X$ and dualize it. We get a map $Df_\ast \underline{\mathbb{Z}}_X\to D\underline{\mathbb{Z}}_Y=\underline{\mathbb{Z}}_Y[m]$. But since we assume $f$ proper, we have $Df_\ast \underline{\mathbb{Z}}_X=f_\ast D\underline{\mathbb{Z}}_X=f_\ast \underline{\mathbb{Z}}_X[n]$, which gives a map $f_\ast\underline{\mathbb{Z}}_X\to \underline{\mathbb{Z}}_Y[m-n]$, which gives a map $H^i(X,\mathbb{Z})\to H^{i+m-n}(Y,\mathbb{Z})$. This is, of course, the map "take the Poincar\'e dual, push forward and take the Poincar\'e dual again".

Another case is when $f$ is a locally trivial fibration with fiber a smooth compact manifold of dimension $k$ (or more generally, a topological submersion). Then, using the adjunction $Hom (f_\ast\underline{\mathbb{Z}}_X,\underline{\mathbb{Z}}_Y)=Hom(\underline{\mathbb{Z}}_X,f^!\underline{\mathbb{Z}}_Y)$, and using the fact that $f^!\underline{\mathbb{Z}}_Y=\underline{\mathbb{Z}}_X[k]$, we see that there is a map $f_\ast\underline{\mathbb{Z}}_X\to \underline{\mathbb{Z}}_Y[-k]$, which gives a map $H^i(X,\mathbb{Z})\to H^{i-k}(Y,\mathbb{Z})$. This is the integration along the fibers map.