Part (a) can be done as Mike suggested in the comments (or using the Liouville theorem for harmonic functions), or considering the function $e^{-if}$ which is bounded by $1$ on the whole plane, hence constant.
For part (b), if $\mathsf{Im}(f(z))$ tends to $0$ as $|z|\to\infty$, then there is $R>0$ such that, for $|z|\geq R$, $|\mathsf{Im}(f(z))|\leq 1$.
Take $K=\{z\in\mathbb{C}\ :\ |z|\leq 2R\}$. This is a compact set, hence $|\mathsf{Im}(f(z))|$ achieves a maximum on $K$, say $M>0$, but outside $K$, $|\mathsf{Im}(f(z))|\leq 1$, so $|\mathsf{Im}(f(z))|\leq\max\{M,1\}$ on the whole plane, hence it is constant.
Equivalently, set $g=e^{-if}$, then $|g|\to1$ when $|z|\to\infty$, meaning that $g$ is bounded around $\infty$. By Riemann's extension theorem, the function $h(z)=g(1/z)$, bounded around $0$, can be extended holomorphically to $z=0$; therefore the function $g$ extends holomorphically (or at least continuously if you have problems dealing with Riemann surfaces) to a function $\tilde{g}:\mathbb{C}\cup\{\infty\}\to\mathbb{C}$, i.e. $\tilde{g}:\mathbb{S}^2\to\mathbb{C}$. But as $\mathbb{S}^2$ is compact, $\tilde{g}$ is bounded and so is $g$, which is then constant and so is $f$.
(or if you know enough about holomorphic functions, on compact Riemann surfaces like $\mathbb{CP}^1=\mathbb{C}\cup\{\infty\}$ there are no non-constant holomorphic functions…)
It means that for some disc $B(a,r)$ we have that
$$ f(\mathbb{C}) \cap B(a,r) = \emptyset$$
Then, for all $z \in \mathbb{C}$,
$$ | f(z) - a | \geq r$$
and you can consider its multiplicative inverse and apply Liouville.
Best Answer
Your argument is correct and the theorem you want to use is Picard's Theorem. But you to complete it by observing that if $u+iv$ attains some value in the disk of radius $\epsilon$ around a real number $a$ then $u$ takes a value in $(a-\epsilon, a+\epsilon)$.
Actually $u$ attains very real value (by continuity). To see this consider $e^{u+iv}$ and $e^{-(u+iv)}$ Using Liouville's Theorem show that $u$ can neither be bounded above nor be bounded below.