Why are copper salts less soluble than zinc salts?
It is because the difference in electronegativity. 
The higher the difference in electronegativity between the cation and the anion, the stronger the ionic bond. 
Since we are comparing species with the same anions and different cations, we only have to look at the values of the latter ones. 

cation    electronegativity   configuration
  Cu             1.9         [Ar] 3d^10 4s^1
  Zn             1.65        [Ar] 3d^10 4s^2


$\ce{Cu}$ is more electronegative so the bonds have less ionic character which makes for stronger bonds. 

The same happens in the same period between $\ce{K}$ and $\ce{Ca}$, but here the situation with regard to the electronic configuration is reverse. 
Potassium salts are much more soluble than calcium salts, and again it is explained by the electronegativity:

cation    electronegativity   configuration
  K              0.82           [Ar] 4s^1
  Ca             1.0            [Ar] 4s^2