Why does the aqueous solution of sodium peroxide turns red litmus into white? A white solid is either $\ce{Na2O}$ or $\ce{Na2O2}$. A piece of red
  litmus paper turns white when it is dipped into a freshly made aqueous
  solution of white solid. 
  
  
  Identify the substance and explain with the help of balanced
  equation.
  Explain what would happen to the red litmus if the white solid
  were the other compound.
  


From the solution and theoretical knowledge, I know the following:

$$\begin{align}
\ce{Na2O2 + 2 H2O &amp;-&gt; 2NaOH + H2O2}\\
\ce{Na2O + H2O &amp;-&gt; 2NaOH}
\end{align}$$ 

The aqueous solution of $\ce{Na2O}$ will convert red litmus into blue (basic property), so, the substance is $\bf{not}$ $\ce{Na2O}$ for sure. 

But, how will it be $\ce{Na2O2}$? Why does the aqueous solution of sodium peroxide turns red litmus into white?
Your analysis seems right and you remember correctly that colour change of litmus of litmus paper will indicate the pH value of the solutions into which it is dipped.

That is, unless the dye (litmus) is irreversible transformed to another compound which no longer shows a pH dependent colour.

As the experiment states: 


  A piece of red litmus paper turns white [...]


With other words: Litmus was bleached.

This bleaching typically is the result of an oxidative transformation of a (former) dye. 

Here, the only conceivable oxidant in solution is hydrogen peroxide. Consequently, the initial compound in question must have been $\ce{Na2O2}$.