Variance of $\hat{\beta _0}$ in case of homoskedasticity Stock and Watson express the variance of $\hat{\beta _0}$ like 
$\hat{\sigma }^2_\hat{\beta _0}=\frac{E({X_{i}}^{2})}{n\sigma _{X}^{2}}\sigma ^{2}$, but starting from variance of $\hat{\beta _1}=\frac{\sigma^{2}}{n\sigma_{X}^{2}}$ i proved only that $\hat{\sigma }^2_\hat{\beta _0}=\frac{1}{n}\sigma^{2}(1+\frac{\bar{X}^2}{\sigma_{X}^{2}})$, that is the same that is showed here. 

How can i prove that are similar forms?
They are certainly not the same, since the first contains an expected value while the second does not contain an expected value but a sample mean.  

Of course, they are very closely connected, and one can see this if one writes down the definition of the variance of a random variable.