Asymptotic runtime of a nested sum
In fact you do change the value of $n$, see $n=n/4$, and so the while-loop results in  $\sum_{i=1}^{\log_4{n}}{\sqrt{i}}$ which is equal to $\sqrt{1} + \sqrt{2} + \dots + \sqrt{\log_4{n}}$.

Upper bound
$\sqrt{1} + \sqrt{2} + \dots +\sqrt{\log{n}} \leq \log{n}\sqrt{\log{n}}$ which is less than $\sqrt{n}$ for sufficiently large $n$s (you may prove it using induction or using some other techniques).