I've solved similar problems and the basic idea is to first prove that $f$ is bounded, meaning we need an inequality of the form $|f(z)|<M$. In some problems the relevant inequality is already given while in others you need to modify the given relation to form it by yourself.
The second and final step is to prove that $f'(z)=0$ by using Cauchy's Integral formula along with the bounding inequality.
In this problem I'm having trouble with the first step. I'm using the $\epsilon - \delta$ definition of the limit which gives: $$\forall{\varepsilon} > 0, \exists{n_0} \in \mathbb{N} : \forall{\left|z\right|} \geq n_0:|e^{f(z)}-4|<\epsilon$$
Now, how do I isolate $|f(z)|$ to get the bounding inequality ?
Observe that $e^f$ is a bounded entire function. Hence it is a constant function, by Liouville's theorem. As the limit of $e^f$ is $4$ as $z\to\infty$, we in turn obtain $e^{f(z)}=4$ for every $z\in\mathbb C$. Therefore, for each $z$, we must have $f(z)=\log 4+2k\pi i$ for some integer $k$ that may possibly depend on $z$. However, since $f$ is continuous, $k$ must remain constant thoughtout $\mathbb C$. Hence $f$ is constant.