Which electron gets which energy level? Electrons sit in different energy levels of an atom, the farther the higher energy is. Every electrons have the same structure, they can gain energy from environment, electrons which gained energy could jump to a higher energy level and will finally fall back again. 
I'm wondering why some electrons have the "right" to "store" that high energy since every electron is the same.  Why do those electrons can have more energy and sit in higher energy level than other electrons?
 A: The electron is not who "wins" energy. The increase in energy corresponds to the system electron-nucleus. The "incoming" energy is stored in the system, by increasing the distance from the nucleus to the electron.
The configuration of the atom, is such that always "looking" the lowest energy state for the system.
A: Pauli's exclusion Principle requires no two electrons to occupy the same quantum state. Based on spin, it is decided which electron 'sits' where it does. As far as the 'jumping' to the higher energy is concerned, it depends on the way the electron gains energy. If say, light of energy which matched the energy difference between two energy level is incident, then the electrons 'jump' to that energy level.
A: I'm surprised that no one has mentioned that there is really no such thing as "this electron" or "that electron" in an atom. Those are useful approximations that help us visualize energy levels; but the actual quantum-mechanical theory of, for example, a carbon atom with six electrons, is based on a single electron wave function in 18-dimensional phase space. 
Or look at a Helium atom if you prefer, with only two electrons. You cannot solve for the wave function of the first excited state by saying "one electron is in the s-state and the other one is in the p-state." You have to write a function in six-dimensional wave space, and it has to be symmetrical in both electrons...so that if you switch "them" around, it's exactly the same function except for a 180 degree phase change.
A: "I'm wondering why some electrons have the "right" to "store" that high energy".
Take a read of energy levels in Bohr's model. Since electron can only revolve in certain orbits they will be at a certain distance from the nucleus. They all will have different kinetic and potential energies.
Electron store different energies because they are having different electrostatic potential(determined by the distance from the nuclues) energy and different kinetic energy(determined by their speed).
your arguments should be the other way round, that is: 

Different electrons have different energies because they sit in different energy level.  


Edit: It should be understood that velocity of an electron revolving in a circular orbit depends upon the radius of the orbit.
Since the motion of electron is considered as circular the acceleration of the electron is constant and can be found easily as:  $F=m_ea$
Also The coloumb's force is $F=\dfrac{Z k_{e} e^2}{r^2}$ 
Also for uniform circular motion $a=v{\dfrac{d\theta }{dt}}=v\omega ={\dfrac  {v^{2}}{r}} $
So $ \dfrac{m_\mathrm{e} v^2}{r} = \dfrac{Zk_\mathrm{e} e^2}{r^2}$
$\implies \dfrac{1}{2}m_ev^2=\dfrac{1}{2} \dfrac{Zk_\mathrm{e} e^2}{r}$   
hence $K.E=\dfrac{1}{2} \dfrac{Zk_\mathrm{e} e^2}{r}$
Note: The signs have their usual meaning. Kinetic energy is calculated for the unrelativistic case i.e. $v<<c$.
A: The electrons are indistinguishable, so whenever you do any real calculation you should treat EVERY electron on equal footing.
So indeed your claim of saying:

I'm wondering why some electrons have the "right" to "store" that high energy since every electron is the same. Why do those electrons can have more energy and sit in higher energy level than other electrons?

Is justified! The correct way of doing this, is by working in Fock-space where every electron is treated the same way.
So as @Tinchito correctly notices, you should look at the SYSTEM, and not at single electrons.
