Note that $ker(f)$ is a sublattice of the integer lattice $\mathbb{Z}^r$. And conversely any such integer sublattice will give you such a homomorphism $f$. The index of this sublattice is exactly what you call $h$ (since the homomorphism is surjective). If $C$ is a symmetric convex body in $\mathbb{R}^r$ and $L$ a full rank lattice then the number of lattice points in $t C$ (the dilation of $C$ by a real factor of $t$) is asymptotic to $t^r/D$, t^r vol(C)/D$, where $D$ is the volume of the fundamental domain of $L$. In your case $D=h$ and $C$ is the unit hypercube. Getting a good remainder is usually a much harder job. In general the remainder is bounded by something proportional to the area of $C$. When $C$ has a smooth boundary (which alas, the hypercube doesn't) you can get better estimates (but usually have to work quite hard). When you're working in a high dimension a large fraction of the volume is close to the vertices -- making things much more difficult.
In particular you might look at the work of Martin Huxley. For example his book "Lattice Points, Area and Exponential Sums" or a number of his papers on this subject.
