Existence of prime ideals and Axiom of Choice.

The existence of prime ideals in commutative rings with unity is equivalent in $ZF$ to the Boolean prime ideal ($BPI$) theorem, which is strictly weaker than the axiom of choice. The first reference for this is D. Scott: "Prime ideal theorems for rings, lattices and Boolean algebras", Bulletin of the American Mathematical Society (60) pp. 390.

As for the relation between BPI and the axiom of countable choice, neither of them implies the other, since there are models of $ZF$ where one holds while the other fails. You can find these in the usual reference, Howard & Rubin: "Consequences of the axiom of choice".

The theorem you mention which implies the existence of prime ideals but seems a bit stronger, is actually equivalent to $BPI$ as well. That it implies $BPI$ is trivial, and the other implication is theorem 4.1 of Rav, Y.: "Variants of Rado's selection lemma and their applications" Mathematische Nachrichten (79) 1, pp. 145.

Although godelian has already answered, let me give a more direct answer (with more proofs instead of references; perhaps they coincide). First, notice that the existence of prime ideals in $R$ disjoint from multiplicative subsets $S$ and containing a given ideal $I$ is actually (quantified over $R$) equivalent to the existence of prime ideals in $R$ (quantified over $R$, of course $\neq 0$). The non-trivial direction just uses $S^{-1} (R/I)$. So we actually have only one statement, the existence of prime ideals.

I claim that the Compactness Theorem (for propositional logic) implies the existence of prime ideals: For each $a \in R$ let $p_a$ be a new variable. Consider the theory whose axioms are $p_0$, $\neg p_1$, $p_a \wedge p_b \longrightarrow p_{a+b}$, $p_a \longrightarrow p_{ab}$, $p_{ab} \longrightarrow p_{a} \vee p_{b}$ for all $a,b \in A$. A model of this theory is precisely a prime ideal of $R$. Since finitely generated rings are noetherian (Hilbert) and noetherian rings have maximal and therefore prime ideals, the theory is finitely consistent. Hence, it is consistent.

On the other hand, the Compactnass Theorem is weaker than the Axiom Choice.

PS: Now I've realized that the whole question is a dublicate of this one. See especially the answer by Chris Phan.

In light of Laurent Moret-Bailly's comment under Martin's answer, let us supplement that answer by showing how existence of prime ideals in finitely generated rings may be proven in ZF without a choice principle (not even the ultrafilter principle). In fact we will show that countable rings $R$ have prime ideals, using the weak König lemma.

(Recall the weak König lemma: if $T$ is an infinite subtree of the infinite binary tree $2^\ast$ [the nodes of $2^\ast$ are finite words in letters $0, 1$; the parent of any nonempty word is obtained by omitting its last letter], then $T$ has an infinite branch. This is easily proven in ZF; see e.g. here.)

Fix an enumeration $a_0, a_1, a_2, \ldots$ of elements of $R$. Build a tree $T$ whose nodes $w$ are labeled by ideals $P_w$ of $R$, by induction. $T_k$ will denote the set of nodes of length $k$, so $T_0$ consists of just the empty word $e$, which we label with the zero ideal. Assuming $T_{k-1}$ and the ideals $P_w$ for words $w \in T_{k-1}$ are given, define $T_k$ and the ideals there according to two cases; for this it is convenient to write $k = 2l + m$ where $m \in \{0, 1\}$ and $l$ is used to code a pair of natural numbers $(i, j)$ by the usual trick, viz. $l = \binom{i+j+1}{2} + j$.

Case $m = 0$: for each $w \in T_{k-1}$ such that $a_i a_j$ belongs to $P_w$, put $w0, w1$ in $T_k$, and define $P_{w0} := P_w + \langle a_i \rangle$, and $P_{w1} := P_w + \langle a_j \rangle$. For any other $w \in T_{k-1}$, put $w0$ in $T_k$, and define $P_{w0} = P_w$.

Case $m = 1$: for each $w \in T_{k-1}$ such that $1 \notin P_w$, put $w0$ in $T_k$, and define $P_{w0} := P_w$. If $1 \in P_w$, then put neither $w0$ nor $w1$ in $T_k$.

$T$ is infinite: it is easy to prove by induction that for each $k$ there is $w \in T_k$ such that $1 \notin P_w$. If $b$ is an infinite branch through $T$, then put $P = \bigcup_{w \in b} P_w$. Then $P$ is a proper ideal by the recipe of case $m=1$, and is prime by the recipe of case $m=0$.