Let $F$ be a non-Archimedean local field, and $\mu_F$ a Haar measure on $F$. The space $C^{\infty}_c(F)$ of locally constant functions of compact support is spanned by characteristic functions of the sets $a+\mathfrak{p}^b$, for $a \in F, b \in \mathbb{Z}$ and $\mathfrak{p}$ the maximal prime ideal of $\mathcal{O}_F$.
I'm trying to prove the following: let $\Phi_0, \Phi_1$ be the characteristic functions of $\mathcal{O}_F, a+\mathfrak{p}^b$, respectively. If \begin{equation} \int_F \Phi_0(x)\, d\mu_F(x)=c_0>0, \end{equation} then \begin{equation} \int_F \Phi_1(x)\, d\mu(x)=c_0q^{-b} \end{equation} where $q$ is the cardinality of the finite field $\mathcal{O}_F/\mathfrak{p}$ and $||x||=q^{-v_F(x)}$ for all $x \in F$. I feel to prove this I must express $\mathcal{O}_F$ in some way related to the sets $a+\mathfrak{p}^b$, and use properties of the Haar measure to find the measure of $a+\mathfrak{p}^b$. Can it be answered like this, or is there another way?
Your idea is correct. Note that by translation invariance, we may assume $a=0$. Now note that $$\mathcal O_F=\bigcup_{x \in \mathcal O_F/\mathfrak{p}^b}x+\mathfrak{p}^b$$ where we implicity chose representatives of $\mathcal O_F/\mathfrak{p}^b$.
By applying translation invariance and additivity, we get that $$\mu(\mathcal O_F)=|\mathcal O_F/\mathfrak{p}^b|\cdot \mu(\mathfrak{p}^b)=q ^b\mu(\mathfrak{p}^b)$$