Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me).
How do I find this integer $\varphi$ with a formula? Is there a closed form solution for this?
Is the solution unique?
Does the solution (closed form or not) have a name?
Clarification. I think this translates as: How many integer radians do I need to rotate anti-clockwise along the unit circle from $(1,0)$ to get to $z$, given that it can be done?
With respect to your last edit: "How many integer radians do I need to rotate anti-clockwise along the unit circle from $(1,0)$ to get to $z$, given that it can be done?"
Given that it can be done, the argument of $z$ is $\arg z = \varphi + 2\pi k$ for a unique pair of integers $\varphi$ and $k$. (I'll leave as an exercise - why is the pair unique?)
Let's change our language in a way that'll help us understand what problem we're dealing with here. Let $G$ be the group $G = \{\left[0, 2\pi\right), +\}$ with the $+$ operation being "addition modulo $2\pi$". Notice that the neutral element here is $0$.
Your problem translates as: "Given an element $z = n\cdot1_G$, how do I recover $n$?"
Note: In an additive group, multiplying by an integer is like taking a power in a "regular" group. So the group-theoretic problem is: Given $a = b^n$, where $a, b$ are known, how do I recover $n$?
This problem is called the discrete logarithm problem. There is no known general formula for the solution of this problem; computationally, it has been compared to the integer factorization problem (especially in how a general fast solution to this problem can break certain cryptographic systems).