Theorem 6 in Chapter XV, $\S$5 of Lang *Algebraic Number Theory* is a result of the sort you want. Lang formulates it using ideles, but he gives an application in more classical language in Example 3 on the following page. I first quote what he shows in Example 3, then I'll try to rewrite it to sound a bit more like what you want.

**Lang's Example 3** Let $k$ be a number field of class number $1$. Let $k_{\infty}$ be the product of the archimedean completions of $k$, so $k_{\infty} \cong \mathbb{R}^s \times \mathbb{C}^t$ if $k$ has $s$ real and $2t$ imaginary embeddings. Let $k_{\infty}^{\ast}$ be the group of invertible elements of $k_{\infty}$, so $k_{\infty}^{\ast} \cong \mathbb{R}^{s+t} \times (\mathbb{Z}/2)^s \times (S^1)^t$, and let $k_{\infty}^{\ast,1}$ be the subgroup of elements with norm $1$. Let $U$ be the unit group of $k$, so $k_{\infty}^{\ast} / U \cong \mathbb{R} \times (S^1)^{s+2t-1} \times \mathbb{Z}/2^j$ for some $j$ and $k_{\infty}^{\ast,1} / U \cong (S^1)^{s+2t-1} \times \mathbb{Z}/2^j$. Let $\sigma : k_{\infty}^{\ast} / U \to S^1$ be a continuous homomorphism whose restriction to $k_{\infty}^{\ast,1} / U$ is surjective. Then $\sigma(\pi)$ is equidistributed in $S^1$, as $\pi$ ranges over generators of prime ideals, each prime ideal taken once.

Okay, but you also want to be allowed to impose congruence conditions on your ideals and work with other class numbers. Lang's Theorem 6 does that, except he only states it idelically. I believe the following is the classical version.

Let $\mathfrak{m}$ be a (nonzero) ideal of $k$ and let $H$ be the $\mathfrak{m}$-ray class group, meaning ideals relatively prime to $\mathfrak{m}$ modulo principal ideals whose generators are $1 \bmod \mathfrak{m}$. By Cebatorov, prime ideals are equidistributed in the finite group $H$, so we can consider a single class of $H$ and ask about the archimedean behavior of primes in that class.

Fix $I$ an ideal in the desired class, so an ideal is in this class if and only if it is of the form $\alpha I$ for some $\alpha \in I^{-1}$ with $\alpha = 1 \bmod \mathfrak{m}$. Let $P \subset I^{-1}$ be the set of $\alpha$ which are $1 \bmod \mathfrak{m}$ for which $\alpha I$ is prime. We have $P \subset I^{-1} \subset k \otimes \mathbb{R} = k_{\infty}$. Roughly, we are going to claim that $P$ is equidistributed.

Let $U$ be the group of units which are $1 \bmod \mathfrak{m}$. Choose a continuous group homomorphism $\sigma$ retracting $k^{\ast}_{\infty}/U$ onto $k^{\ast,1}_{\infty}/U$. Then $\sigma(P)$ is equidistributed in the compact group $k^{\ast,1}_{\infty}/U$.

If this is too abstract, note that a concrete case is equidistribution of Gaussian primes in wedge shaped regions of $\mathbb{C}$. See here, which is where I learned of the Lang reference.