Is Fodor's Lemma necessary for the $\omega_1$ train station puzzle?

The following theorem is due to Neumer and is provable in $\sf ZF$:

Suppose that $\operatorname{cf}(\alpha)>\omega$, if $f\colon\alpha\to\alpha$ is a regressive function, then there is some $\beta<\alpha$ and an unbounded set $A$ such that $f(a)<\beta$ for all $a\in A$.

So if $\omega_1$ is regular, and $f$ is a regressive function, there is a countable ordinal such that unboundedly many points are mapped below it. But since $\omega_1$ is regular, we can partition this unbounded set to the various fibers, and one of them will have to be unbounded. Therefore, the weak version of Fodor's lemma holds whenever $\omega_1$ is regular.

Of course, if $\omega_1$ is singular, which is of course consistent with $\sf ZF$, then the above is moot and by fixing a cofinal sequence $\alpha_n$ for $n<\omega$, we can define a regressive function $\alpha\mapsto\min\{n\mid\alpha<\alpha_n\}$, which is not constant on any unbounded set (we may need to assume that $\alpha_0=0$ and $\alpha_1=\omega$, but that's fine). Now, if $\omega_1$ is singular indeed, then we can easily arrange for the train to arrive with as many passengers we want (none, finitely many, countably many, or even $\aleph_1$ of them if you allow "countably many" rather than explicitly enumerated countable set of passengers!), and therefore the ticketmaster at $S_{\omega_1}$ are no longer safely taking a nap.

To your question, then, yes, the emptiness of the train at $S_{\omega_1}$ is equivalent to the weak Fodor's lemma.