In the beginning of this link:
https://www.math.ku.edu/~lerner/GR/Schwarzschild.pdf
they calculate the acceleration of a stationary observer. As I understand, this acceleration is seen by an omnipotent observer (of the Schwarzschild metric). How can this be the acceleration felt by the stationary observer himself? In addition, what is a metric of the stationary frame? (obviously it's not the omnipotent observer's metric, or it is?)
The acceleration you describe is the four-acceleration, and it's calculated using the geodesic equation. The four-acceleration is an invarient, however to write it down we need to choose a set of coordinates, and the representation of the four-acceleration will look different in different coordinate systems. The stationary observer you describe is usually called a shell observer.
In GR the way to detect spacetime curvature is to surround yourself with a sphere of test masses and see what happens to them. If you are accelerating the sphere will accelerate away from you. Though not relevant here the sphere also detects tidal forces by changing shape and a non-zero Ricci tensor by changing volume.
The metric is an invariant, like the four-acceleration, but it too will look different when written down in different coordinate systems. The Schwarzschild metric we all know and love is written in Schwarzschild coordinates, that is the $t$, $r$, $\theta$ and $\phi$ are the coordinates as measured by an observer an infinite distance, and obviously the metric will look different to a shell observer. I don't know how to write the metric for the shell observer, but locally it will look like a Rindler metric. If you're interested, my answer to Curvature gravity and a falling apple? shows how to calculate the acceleration for a Rindler observer.
Response to comment:
You ask:
Is writing down the metric for a shell observer impossible?
To answer this involves one of the concepts that beginners to GR often find hard. It's a key principle in GR that we can use any coordinates we want, so the shell observer can simply choose to use the Schwarzschild coordinates and the metric is just the Schwarzschild metric. But if the shell observer does this then the $r$ coordinate wouldn't be the same as the $r$ the observer measures with their ruler.
When we use phrases like in the shell observer's coordinates we typically mean in coordinates that look locally to the shell observer like flat space coordinates. We might also ask that the origin of the coordinates be at the shell observer's position (though probably not in this case). Suppose we define new coordinates:
$$ dr_{shell} = \frac{dr}{\sqrt{1 - GM/r}} $$
$$ dt_{shell} = dt\sqrt{1 - GM/r} $$
The $dr_{shell}$ and $dt_{shell}$ do match measurements made with the shell observers rulers and clocks, so they seem a reasonable choice. If we write the metric in these coordinates we get:
$$ ds^2 = -dt_{shell}^2 + dr_{shell}^2 + r^2d\Omega^2 $$
Which actually looks like flat spacetime. But it only looks flat because that's how we chose $dr_{shell}$ and $dt_{shell}$. The metric also still contains the Schwarzschild $r$ coordinate in the $r^2d\Omega^2$ term. It's certainly possible to rewrite $r$ as a function of $r_{shell}$ instead, though I fear it would be a messy function and offhand I don't know what this function is.
