How can Nernst equation apply in this redox reaction? $$\ce{CuSO4 + Zn -&gt; ZnSO4 + Cu}$$
What is the cell potential of the voltaic cell with copper and zinc electrodes if the system is at 50 degrees and the solutions are both at 1 mol? How can it be equal to the standard potential from the nernst equation? $$E_\text{cell} = E^\circ – \frac{RT}{nF}\ln Q$$

If the solutions are at 1 mol concentrations each then $Q = 1$ and $\ln Q = 0$ resulting in $E_\text{cell} = E^\circ$.
But the above basically shows that concentration affects cell voltage but temperature doesn't have an effect on cell potential – which isn't the case. How can I calculate how different temperatures affect the cell voltage?
The flaw in your reasoning is that you are believing that $E^\circ$ is a constant, when really it is a function of temperature.  

See Standard Electrode Potentials and Temperature Coefficients in Water at 298.15K for a linear approximation of the temperature dependence of $E^\circ$.