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 $\ce{Na2O2}$.