A co-meagre topology on $\aleph_{\omega}$

Your intuition was correct, the space you describe ist not first countable. Note that your question is equivalent to the following:

Let $S=\{a\subseteq\aleph_\omega\mid \aleph_\omega\setminus a\in\mathcal P_{\aleph_\omega}(\aleph_\omega)\}$. Does $(S, \supseteq)$ have a countable dense subset?

I will argue that the answer is no. Suppose $D$ is such a set. Wlog, we may assume that $D=\{d^n_m\mid n,m<\omega\}$ where $c^n_m:=\aleph_\omega\setminus d^n_m$ has size $\aleph_n$. As $\aleph_{n+1}$ is regular we have that $(\bigcup_{m<\omega} c^n_m)\cap\aleph_{n+1}$ is bounded in $\aleph_{n+1}$, say it is bounded by $\beta^n<\aleph_{n+1}$. Now $a=\aleph_\omega\setminus\{\beta^n\mid n<\omega\}$ provides a counterexample to the density of $C$: For any $n, m<\omega$, we have that $\beta^n\in d^n_m\setminus a$!

A slight modification of this argument shows that in fact, $(S, \supseteq)$ does not have a dense subset of size ${<}\aleph_\omega$.So I must say, this space is a really good example for your exercise: Not only does no point have a local base which is countable, no point has a local base of size smaller than $\aleph_\omega$!