Electrons in an atom have quantized energy quantity. Can uncertainty principle be applied in this case, then?
How does this work?
As energy is fixed, this seems to disobey $\Delta E \Delta t \geq \hbar/2$...
You can only measure the energy of an electron in an atom perfectly if you measure it for an infinitely long time. If you do your measurement for a finite time there will be a finite uncertainty due to the uncertainty principle.
This is not some esoteric piece of mathematics, it's a real and measurable effect. For example if you measure the emission spectrum from an atom you are measuring the difference in energy between some excited state and a lower energy state. Because the lifetime of the excited state is short it's energy is uncertain, and consequently the energy of the emitted photon is uncertain. The result is that the lines in the emission spectrum are not infinitely sharp. They have a finite width due to the uncertainty principle.