How did Bohr come up with the figure $n\hbar$? Bohr gave his atomic model in the year 1913, whereas De Broglie proposed his idea of matter waves in the year 1924 (from which the relationship can be derived by considering an electron to be a standing wave). How is it then that Bohr came up with this postulate?
 A: This is a summary of the Wikipedia Bohr Model  article.
The Bohr model was developed  without either the  knowledge or need of De Broglie's later work. 
In 1912 Bohr was aware of the problems with the Rutherford "solar system" model:
 
Rutherford Model's Problems


*

*The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus.

*Because the electron would lose energy, it would rapidly spiral inwards, collapsing into the nucleus on a timescale of around 16 picoseconds.

*This atom model is disastrous, because it predicts that all atoms are unstable.

*Also, as the electron spirals inward, the emission would rapidly increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. 
The key  to Bohr's insights may have been that, instead of a continous spectrum, atoms were discovered with discrete emission energies.   
Bohr proposed, in 1913, that  electrons could only have certain classical properties. Bohr supported his contention  by appealing to the correspondence principle, without providing any sort of wave interpretation. In 1913, the wave behavior of matter particles such as the electron (i.e., matter waves) was not suspected.

Bohr Model
The Bohr model was based upon Planck's quantum theory of radiation. Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency $ν$:
$${\displaystyle \Delta {E}=E_{2}-E_{1}=h\nu \ ,}$$
where $h$ is Planck's constant. The frequency of the radiation emitted at an orbit of period $T$ is as it would be in classical mechanics; it is the reciprocal of the classical orbit period:
$${\displaystyle \nu ={1 \over T}~~.}$$
The importance of the Bohr model is that we could utilise the laws of classical mechanics to the electron's motion, but modified  by a quantum rule. Although his model was  not completely well defined for small orbits, because the emission process involves two orbits with two different periods, Bohr could determine the energy spacing between levels  and come to an exactly correct quantum rule: the angular momentum $L$ is restricted to be an integer multiple of a fixed unit:
$${\displaystyle L=n{h \over 2\pi }=n\hbar }$$
where $n = 1, 2, 3, ...$ is called the principal quantum number, and $ħ = h/2π$.  n has a lowest value of 1; this gives a smallest possible orbital radius of 0.0529 nm known as the Bohr radius. Once an electron is in this lowest orbit, it will not "drop" any further into the nucleus. Starting from the angular momentum quantum rule, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.

Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted in 1924 by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

$${\displaystyle n\lambda =2\pi r~.}$$
Substituting de Broglie's wavelength of $λ = h/p$ reproduces Bohr's rule.
