Is it possible to calculate energy of electron in any orbital and atom in Schrödinger wave model theory?
It is surely possible. Note though, that for many-electron systems orbital picture is an approximate description, but that is an approximation on which the whole general chemistry is based on, so that is fine from that perspective. The approximation is known as the Hartree-Fock (HF) model, and thus, to get the energies of electrons occupying some orbitals all you need to do is to solve the Schrödinger equation in this approximation.



As a quick exercise one can do HF calculation in, say, Gaussian, with the following input file:

#P HF/aug-cc-pVTZ Pop=Full

Na-

-1 1
 Na                 0.00000000    0.00000000    0.00000000


to get this picture of occupied orbitals:



The HOMO with the energy of $-0.01288 \, \mathrm{Hartree} = -0.3505 \, \mathrm{eV}$ is basically the $\mathrm{3s}$ orbital you're looking for (trust me). Note, however, that this energy is approximate since the basis set (aug-cc-pVTZ) is finite. We could do better than that, but that is a different story.



In response to permeakra (since he does not trust his colleagues) I quote an authoritative reference in the field which explicitly talks about individual electrons occupying individual spin-orbitals and orbital energies. As a reference I choose the infamous book entitled "Modern Quantum Chemistry" written by Attila Szabo and Neil S. Ostlund.

Quote #1 (Szabo &amp; Ostlund, p. 50)


  This Slater determinant has $N$ electrons occupying $N$ spin orbitals
  $(\chi_i, \chi_j, \dotsc, \chi_k)$ without specifying which electron
  is in which orbital.


Quote #2 (Szabo &amp; Ostlund, p. 54)


  $$   f(i) \chi(x_i) = \varepsilon \chi(x_i) \tag{2.52} $$ [...]  
  
  The
  solution of the Hartree-Fock eigenvalue problem (2.52) yields a set 
  $\{\chi_k\}$ of orthonormal Hartree-Fock spin orbitals with orbital
  energies $\{\varepsilon_k\}$.


Below I also quote what Szabo &amp; Ostlund have to say about Koopmans' theorem, because I already know that won't trust me.


  Quote #3 (Szabo &amp; Ostlund, p. 110)
  
  The first theorem [Koopmans' theorem] constitutes an interpretation of
  the Hartree-Fock orbital energies as ionization potentials and 
  electron affinities.


See, orbitals with their energies &amp; occupancies do exist in the Hartree-Fock theory. It is the interpretation of orbital energies as ionization potentials which requires an additional theorem.