Electron's Energy In The Atom
Can we explain an electron's energy of a specific orbital using the
  Penetration and Shielding effect?


Yes, to some extent it is possible. Look no further than Slater's rules which specifies the values of the effective nuclear charge $Z^*$, and the effective principal quantum number $n^*$ that can be used to replace the nuclear charge $Z$ and the principal quantum number $n$ in the hydrogen-like atom orbital energy expression. The energy of each and every orbital can be approximated then by
$$
    E = - \left( \frac{Z^*}{n^{*}} \right)^{2}. 
$$
But I would be careful calling this an explanation, since it is rather a rationalization. I wrote about the difference between those two here.