Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at a prime $p$. It is well-known that the map $$E[N]\to E_p[N]$$ is injective when $p\nmid N$. It is even a bijection since both groups have the same size.
Is anything known when $p\mid N$ ? All the books I've consulted about elliptic curves only consider the case $p\nmid N$.
For example, if $E$ has ordinary reduction, we have that $E[p]\cong(\mathbb{Z}/p\mathbb{Z})^2$ and $E_p[p]\cong \mathbb{Z}/p\mathbb{Z}$, so we could maybe expect a surjection from $E[p]$ onto $E_p[p]$...
First note that when you say that the reduction map on torsion is isomorphism (if $p \not\mid N$) you have to pass to some suitably large extension of $\mathbb Q$, since typically the $N$-torsion points are not defined over $\mathbb Q$.
In general (i.e. even if $p \mid N$), if you pass to a suitably large extension of $\mathbb Q$ (depending on $N$), then the reduction map on $N$-torsion will be surjective. (Using the Chinese Remainder Theorem, one easily reduces to the two cases when $p \not\mid N$ or $N$ is a power of $p$. The first case you've already discussed. The second case can be dealt with e.g. using the point of view of finite flat group schemes, although there is presumably a more elementary treatment; if this is not discussed in the discussion of formal groups in Silverman, then you could try looking in Serre's paper on points of finite order on elliptic curves.)