Let $\mathcal{L}$ be a countable language, and let $\mathcal{A}$ be an $\mathcal{L}$-structure with universe $ω_1$.
Let $C$ be the set of countable ordinals $α$ such that the restriction of $\mathcal{A}$ to the set $α = \{β : β < α\}$ is an elementary substructure of $\mathcal{A}$. Then $C$ is a club. (I proved this part).
Now, in the structure $\mathcal{A}$, the elements are countable ordinals. Suppose that for each $α ∈ C$, there is some $β < α$ such that $\operatorname{tp}_{\mathcal{A}}(α)$ = $\operatorname{tp}_{\mathcal{A}}(β)$.
Show that some fixed type is realized by uncountably many elements.
I have an inkling that this should follow from a well-known theorem, but don't know what the theorem is and how the proof might go.
The well-known theorem you're looking for is Fodor's lemma.