Non-trivial natural endomorphisms on $\operatorname{Top}$
Solution 1:
Yes, this is correct. More directly, you can just consider any space $X$, any $x\in X$, and the map $*\to X$ which sends the point to $x$. Naturality of $\eta$ with respect to this map says exactly that $\eta_X(x)=x$. (This argument is really exactly the same as yours, though, when you unwind the proof of Yoneda's lemma.)