The Haar measure on the $p$-adic integers is a measure which is translation invariant, finite for compact sets and positive for sets with non-empty interior. As such we can define it so that the Haar measure of the whole set of $p$-adic integers $\mathbb{Z}_p$ equals $1$.
Does there exist a nice proof for the fact that the Haar measure of $\{w\in\mathbb{Z}_p\mid \lvert w\rvert\leq 1/p^n\}= N_n/p^n$, where $N_n = \#\{x\mod p^n\mid x\in\mathbb{Z}_p, f(x)=0\}$ with $f\in\mathbb{Z}_p[x]$?
No idea what is your $f$. If $\mu(Z_p)=1$ then for $n\ge 0$ $$p^n \mu(a+p^n Z_p) = \sum_{b=0}^{p^n-1}\mu(b+p^n Z_p)=\mu(\bigcup_{b=0}^{p^n-1}b+p^n Z_p)=\mu(Z_p)=1$$ $p^n Z_p=\{ c\in Z_p,|c|_p\le p^{-n}\}$.