I know the radius $R$ of the circle and the area $A$ of the segment.
How can I solve for central angle $\alpha^{\circ}$ in this (or some other) equation:
$$A=\frac{R^{2}}{2} \left( \frac{\alpha \pi}{180}-\sin \alpha^{\circ} \right)$$
?
Here Newton's algorithm is recommended, but with an initial guess of
$$x(0) = (6k)^{1/3}$$
Why is this the initial guess?
Your equation is transcendental,closed form solution is not possible. Newton -Raphson numerical iteration method is often used. If an approximate solution is acceptable, a graphical solution is also one method.
EDIT1:
By series expansion upto 2 terms we get a good approximation
$$ 2 A /R^2 = k \approx \alpha - \sin \alpha = \alpha ^3 /6$$
so we can choose a reasonably accurate value for starting iteration as:
$$ \alpha_{initial}= (6 k)^ { \frac13} .$$