In this other question on mathoverflow, Eric Wofsey says that for any topological space $X$, the maximal ideals of $C(X)$ correspond to the points of the Stone-Cech compactification $\beta X$. He then says that if $C(X)/I$ is isomorphic to $\mathbb{C}$, then every continuous function on $X$ extends continuously to that point in $\beta X$. My intuition is that you'll get what you want if you can construct a proper continuous function from $X$ to the real numbers; as usual proper means that the inverse image of any compact set is compact. I don't know that you would need conditions on $Y$. I also don't know whether your conditions on $X$ yield such a function, but they look similar.

(This is not meant as a complete answer, but it is something.)

In this other question on mathoverflow, Eric Wofsey says that for any topological space $X$, the maximal ideals of $C(X)$ correspond to the points of the Stone-Cech compactification $\beta X$. He then says that if $C(X)/I$ is isomorphic to $\mathbb{C}$, then every continuous function on $X$ extends continuously to that point in $\beta X$. My intuition is that you'll get what you want if you can construct a proper continuous function from $X$ to the real numbers; as usual proper means that the inverse image of any compact set is compact. I don't know that you would need conditions on $Y$. I also don't know whether your conditions on $X$ yield such a function, but they look similar.

(This is not meant as a complete answer, but it is something.)