Random Walk Upper-Bound. For any symmetric Random Walk, absolute value of partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}$? Is it possible to prove that for any symmetric Random Walk, the absolute value of its partial sum $S_n$ never exceeds $\sqrt{2 \pi n}+\sqrt{\frac{\pi}2}\quad\large?$

I run some simulation, clearly for initial value it is not true, but for large $n$ it seems true that it is a real absolute bound.

No 
For example you might with positive probability get nine $+1$s in a row.
But $\sqrt{18 \pi}+\sqrt{\pi / 2} \lt 9$.
More substantially, the law of the iterated logarithm says $\displaystyle \limsup_{n \to \infty} \frac{S_n}{\sqrt{2n \log\log n}} = 1$ almost surely. For big enough $n$ you will have $\dfrac{\sqrt{2n \log\log n}}{\sqrt{2n \pi} + \sqrt{\pi/2}} \gt 1$. 
