Ultrafilters on naturals determine p-adic numbers

The set $\beta(\mathbb{N})$ has a natural topology that makes it the Stone-Cech compactification of $\mathbb{N}$, and your map $\Phi$ is just the unique continuous extension of the inclusion map $\mathbb{N}\to\hat{\mathbb{Z}}$. In particular, since $\Phi$ is continuous, its image is closed in $\hat{\mathbb{Z}}$. Since $\mathbb{N}$ is dense in $\hat{\mathbb{Z}}$, this implies $\Phi$ is surjective.

Concretely, given $a\in\hat{\mathbb{Z}}$ and nonzero $n\in\mathbb{Z}$, let $X_{a,n}$ be the set of integers that have the same mod $n$ residue as $a$. For fixed $a$, these sets $X_{a,n}$ generate a proper filter (since $X_{a,n}\cap X_{a,m}=X_{a,\operatorname{lcm}(m,n)}$) which can then be extended to an ultrafilter, and this ultrafilter maps to $a$ under $\Phi$.