How to explain the excited states in the dinitrogen cation?
Excited states make things a bit more complicated. The electron configuration only represent the HF ground state, and changing the occupied orbitals will have an effect on all the orbitals (they are correlated). So this picture becomes more and more inaccurate the higher lying the electronic state. What you do to resolve this is expand the electronic state in many configurations (which all most belong to the same IRREP).

For diatomic molecules there is a lot of spectroscopic data available. So let's check it: $\ce{N2+}$ in the NIST Chemistry WebBook

The states in that table are ordered by energy, with the ground state at the bottom. So scrolling down shows us the 3 lowest lying states: $\rm X ^2\Sigma_g^+$, $\rm A^2\Pi_u$, $\rm B^2\Sigma_u^+$.

Basically, this means you are right about the first state. To explain why we need a bit of symmetry. The first excited state has $\Pi_u$ symmetry, in the $D_{\infty h}$. Since we are working with $D_{2h}$ here, we will need to reduce it. Sparing the details, the result is $\Pi_u$ becomes $B_{2u}\oplus B_{3u}$, which means it will show up once in each IRREP, which is fine because $\Pi$ states are doubly degenerate anyway (just as $\pi$ orbitals are doubly degenerate).

The other state ($\Sigma_u^+$) becomes $B_{1u}$. This one you can for example get by excitation from the anti-bonding $2\sigma_u$ into $3\sigma_g$, with respect to the ground state configuration (see this reference). So there is an electron "jumping over" the $1\pi_u$ orbitals.

Note, that the ordering of the states can change if you start to move away from the equilibrium distance towards dissociation.