Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this compactification (or equivalently, $x$ and $y$ are two non-principal ultrafilters). I am interested in ways to "model" the ultrafilter $y$ using the ultrafilter $x$. More precisely,

Q1. (Existence) Does there necessarily exist a continuous map $f: \beta \mathbb{N} \to \beta \mathbb{N}$ which maps $x$ to $y$, while mapping $\mathbb{N}$ to $\mathbb{N}$? To put it another way: does there exist a function $f: \mathbb{N} \to \mathbb{N}$ such that $\lim_{n \to x} f(n) = y$?

Q2. (Uniqueness) Suppose there are two continuous maps $f, g: \beta \mathbb{N} \to \beta\mathbb{N}$ with $f(x)=g(x)=y$, which map $\mathbb{N}$ to $\mathbb{N}$. Is it then true that $f$ and $g$ must then be equal on a neighbourhood of $x$?

I suspect the answer to both questions is either "no" or "undecidable in ZFC", though perhaps there exist "universal" ultrafilters $x$ for which the answers become yes. But I do not have enough intuition regarding the topology of $\beta \mathbb{N}$ (other than that it is somewhat pathological) to make this more precise. (The fact that $\beta\mathbb{N}$ is not first countable does seem to indicate that the answers should be negative, though.)