Voltaic cell and Nernst equation
A correct standard notation for the galvanic cell would also include the physical state of each species involved. As such, the correct notation for the given cell is:

$$\ce{Cu(s)}~|~\ce{CuSO4(aq)}~(10^{-3}~\mathrm{M})~||~\ce{CH3COONa~(aq)}~(10^{-2}~\mathrm{M})~|~\ce{Pt(s)}~(\ce{H2(g)})~(1~ \mathrm{atm})$$

Recall that aqueous solutions of salts are generally accompanied by their molarity in solution. Also, metals are usually solid, while gaseous species are accompanied by a reference electrode (platinum in this case) and their pressure value. 

In your case, notice that the ionic salt $\ce{CuSO4}$ is assumed to be a $100\%$ dissociated ionic salt, hence, its aqueous solution will produce $[\ce{Cu^{2+}}]=[\ce{SO4^{2-}}]=10^{-3}\mathrm{M}$. That should answer your final query.

Apart from that, your expression for $\mathrm{Q_c}$ is wrong. It should be $\displaystyle\mathrm{Q}=\frac{p_{\ce{H2}}}{[\ce{Cu^{+2}}]}$. Since $\ce{H2}$ is present in gaseous form, use its pressure value in the expression for $\mathrm{Q}$  (and not molarity which doesn't exist)

PS: The concentration ($\pu{10^−3M}$) that's given to you is the concentration of copper ions after equilibrium has been achieved for the interconversion of $\ce{Cu+2(aq)}$ and $\ce{Cu(s)}$.