The paper in question is now on the arXiv here: "Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole". This gives the following orbital elements (Table E.1):
a = 125.058 mas
e = 0.884649
i = 134.567°
ω = 66.263°
Ω = 228.171°
The orbital elements $i$, $\omega$ and $\Omega$ are essentially Euler angles describing the orbit's orientation in space. Start with the orbit in the $xy$-plane, with the periapse pointing towards $+x$. Then rotate about the $z$-axis by $\omega$, rotate about the $x$-axis by $i$, then rotate about the $z$-axis again by $\Omega$.
The usual convention I've seen is to have the $x$-axis pointing North (positive declination) and the $y$-axis pointing Eastwards (positive right ascension).
Putting it all together, here's some Python code to plot the orbit:
import numpy as np
import matplotlib.pyplot as plt
N = 1800
# generate eccentric anomaly values
Evals = np.linspace(-np.pi, np.pi, N*2+1)
# orbital elements
a = 125.058
e = 0.884649
i = np.radians(134.567)
omega = np.radians(66.263)
Omega = np.radians(228.171)
b = a * np.sqrt(1-e*e)
# orbit in xy-plane with periastron towards +x
xvals = a*(np.cos(Evals) - e)
yvals = b*np.sin(Evals)
# rotate about z-axis by ω
xvals2 = xvals*np.cos(omega) - yvals*np.sin(omega)
yvals2 = xvals*np.sin(omega) + yvals*np.cos(omega)
# rotate about x-axis by i
xvals3 = xvals2
yvals3 = yvals2*np.cos(i)
# rotate about z-axis by Ω
xvals4 = xvals3*np.cos(Omega) - yvals3*np.sin(Omega)
yvals4 = xvals3*np.sin(Omega) + yvals3*np.cos(Omega)
# plot the orbit - note that y is RA and x is Dec
plt.plot(yvals4, xvals4)
# plot the black hole
plt.plot(0, 0, marker='o')
# plot the position of pericentre
plt.plot(yvals4[N], xvals4[N], marker='o')
# plot the line of apsides
plt.plot([yvals4[0], yvals4[N]], [xvals4[0], xvals4[N]], linestyle='--')
# plot the closest point in projected separation
proj_sep = np.sqrt(xvals4*xvals4 + yvals4*yvals4)
min_args = np.argmin(proj_sep)
plt.plot(yvals4[min_args], xvals4[min_args], marker='o')
# RA increases to the left
plt.gca().invert_xaxis()
plt.gca().set_aspect('equal', adjustable='box')
plt.xlabel('ΔRA (mas)')
plt.ylabel('ΔDec (mas)')
plt.show()
This results in the following sky-projected orbit, where the orange dot is the black hole, the green dot is the position at pericentre, and the purple dot is the position at closest projected separation:

This matches the shape of the orbit shown in the paper's Figure 1. Now to compare that to the video screenshot, my best effort at matching the displayed orbits gives the following:

While not a perfect match, it looks as though they might be depicting the closest approach in projected separation, rather than the closest approach in 3D space.