How to distinguish first excited state and second excited state?
Unfortunately, no. If you don't want to work it out by hand, the best you can hope for is that the authors explicitly specify orbital occupations with something that looks like an electron configuration. This paper doesn't, the previous paper sort of does. 

Here is a paper that doesn't have symmetry labels, but does have configurations. For example, on page 2977, the SCF reference for $\ce{CuO}$ is given as

$$
\ce{ (O 2s)^2 (O 2p\sigma)^1 (O 2p\pi)^4 (Cu 3d\sigma)^2 (Cu 3d\pi)^4 (Cu 3d\delta)^4 },
$$

where the wavefunction symmetry is determined by the singly-occupied orbital (everything else is doubly occupied). $\ce{2p\sigma}$ probably lies along the bond, which would be taken as the $z$-axis when looking at a character table. Go to the $C_{\infty v}$ table, check for $z$, and discover that it's $\Sigma^+$, which is even mentioned in the paper. So there are symmetry labels given here, but the linear groups also have Greek characters for their irrep labels...

One interesting point is that non-Abelian groups are not used in practical quantum chemistry. $C_{\infty v}$ would be reduced to $C_{2v}$, and the $^{2}\Sigma^+$ state would become $^{2}\!A_{1}$.