Does Fodor's lemma fail for countable ordinals?

For the usual statement of Fodor's lemma to make sense, one needs well-behaved notions of club and stationary sets, which fail for countable ordinals, so let me be more precise.

**Question:** Let $\alpha$ be a countable limit ordinal. Does there exist a weakly increasing, regressive function $f: \alpha \to \alpha$ such that the image of $f$ is cofinal in $\alpha$?

Here "regressive" means "$f(\beta) < \beta$ for all $0 < \beta < \alpha$", and "weakly increasing" means "$f(\beta) \leq f(\gamma)$ for all $\beta \leq \gamma <\alpha$". I'm also interested in the version of the question where $f$ is defined only on limit ordinals $<\alpha$.

**Motivation:** For me, the biggest significance of Fodor's lemma is the following. Whenever I'm trying to construct something by transfinite induction, and I find myself saying "well, I can't do the construction at this step directly, but if I just go back at this stage and modify the results of a few of the previous steps a little bit, then ..." I immediately stop, because if I keep doing this, then by Fodor's lemma my construction will never "settle down": there will be necessarily be stages of the construction which keep being modified at later and later stages, and the whole thing will never be well-defined. However, if Fodor's lemma fails for countable ordinals, then there's some possibility that a strategy of this form can be successful for constructions which induct only over a countable ordinal. One just needs to take a function $f$ as above, and guarantee that at stage $\beta$, one does not modify anything which came before stage $f(\beta)$.