$\def\mr{\mathrm}$As it happens, quantifier elimination for Skolem arithmetic came up recently in my research. The concise description is that every formula $\phi(x_1,\dots,x_k)$ is in $(\mathbb N^{>0},{\cdot})$ equivalent to a Boolean combination of formulas expressing
$$\tag1\bigl|\{p\in\mathbb P:\psi(v_p(x_1),\dots,v_p(x_k))\}\bigr|\ge n,$$
where $\psi(y_1,\dots,y_k)$ is a formula of Presburger arithmetic, and $n\in\mathbb N$.

In the special case of formulas in one variable with parameters that you are interested in, this boils down to the following: definable subsets are Boolean combinations of sets defined by

$v_q(x)=n$,

$v_q(x)\equiv a\pmod m$,

$\bigl|\{p\in\mathbb P:v_p(x)=n\}\bigr|\ge b$,

$\bigl|\{p\in\mathbb P:v_p(x)\ge n,v_p(x)\equiv a\pmod m\}\bigr|\ge b$,

for some $q\in\mathbb P$, $n,b\in\omega$, $0\le a<m<\omega$.

That all definable relations in $(\mathbb N^{>0},{\cdot})$ are equivalent to Boolean combinations of (1) follows from the results of Mostowski [1]. I will sketch how to prove the other direction, that all sets of the form (1) are first-order definable.

Using $\cdot$, we can define the divisibility, coprimeness, and primality predicates as
$$\begin{align*}
x\mid y&\iff\exists z\,(y=x\cdot z),\\
x\perp y&\iff\forall z\,(z\mid x\land z\mid y\to z=1),\\
\mr{Prime}(x)&\iff x\ne1\land\forall z\,(z\mid x\to z=1\lor z=x).
\end{align*}$$
Then, we can define the set of powers of a prime by
$$\mr{Power}(p,x)\iff\mr{Prime}(p)\land\forall z\,(z\perp p\to z\perp x).$$
Finally, we can define for a given $x$ and a prime $p$ the power of $p$ that appears in the factorization of $x$ by
$$\mr{Val}(p,x,y)\iff\mr{Power}(p,y)\land\exists z\,(x=y\cdot z\land z\perp p).$$
Now, for each prime $p$, $(\{x:\mr{Power}(p,x)\},{\cdot})$ is a model of Presburger arithmetic (which I assume to be formulated in a language with just a single binary function symbol $+$). Thus, if $\psi(y_1,\dots,y_k)$ is a formula of Presburger arithmetic, let $\psi^p(y_1,\dots,y_k)$ (with an extra free variable $p$) denote the formula of Skolem arithmetic obtained by replacing all occurrences of $+$ with $\cdot$, and relativizing all quantifiers to $\{x:\mr{Power}(p,x)\}$. Then (1) is defined by the formula
$$\exists^{\ge n}p\,(\mr{Prime}(p)\land\exists y_1,\dots,y_k\,(\mr{Val}(p,x_1,y_1)\land\dots\land\mr{Val}(p,x_k,y_k)\land\psi^p(y_1,\dots,y_k))).$$

EDIT: I defined $\psi^p$ for formulas written in the language with $+$ only to keep the definition succinct, but in practice, it is more convenient to define it directly for a richer language: specifically, we may translate the constants $0$ and $1$ to $1$ and $p$, respectively, and $x\le y$ to $x\mid y$.

To put it differently, any Presburger formula $\psi(\vec y)$ is equivalent to a Boolean combination of integer inequalities $n+\sum_{i<k}n_iy_i\le m+\sum_{i<k}m_iy_i$, and congruences $y_i\equiv a\pmod m$. We may translate the former to $p^n\prod_{i<k}y_i^{n_i}\mid p^m\prod_{i<k}y_i^{m_i}$, and the latter to $\exists z\,(y_i=p^az^m)$.

**Reference:**

[1] Andrzej Mostowski, *On direct products of theories*, Journal of Symbolic Logic 17 (1952), no. 1, pp. 1–31.