Maximal ideal space of $c_{\mathcal{U}}$

(I will use $\omega$ for the ultrafilter since using a lowercase letter will improve readability)

The way I see it, your algebra $c_\omega$ is simply $$ c_\omega=\{f:\ f(\omega)=0\}\subset C(\beta \mathbb N). $$ So you can make the identification $c_\omega=C_0(\beta\mathbb N\setminus\{\omega\})$.

Note that $c_\omega$ is an ideal in $C(\beta\mathbb N)$, so the ideals in $c_\omega$ are ideals in $C(\beta\mathbb N)$. This is important because, with $\beta\mathbb N$ being compact, the ideals of $C(\beta\mathbb N)$ are precisely the sets of functions that annihilate a fixed closed subset.

Thus, the ideals of $c_\omega$ are the sets of the form $$ \{f\in C_0(\beta\mathbb N\setminus\{\omega\}):\ f=0\ \mbox{ on }\{\omega\}\cup K\} $$ for a fixed closed $K\subset\mathbb N\setminus\{\omega\}$.

We conclude that maximal ideals of $c_\omega$ are of the form $$ \{f\in C_0(\beta\mathbb N\setminus\{\omega\}):\ f(\omega)=f(\eta)=0\} $$ for some $\eta\in\beta\mathbb N\setminus\{\omega\}$.