Neutrino Cross-section approximation: First, thank you for taking the time to read this question.
In section 5 of this paper  the author explained that for energies between  $1-20$ GeV the total Charged-Current Cross sections for $\nu_{\mu}(\bar{\nu}_\mu)$ can be approximated as a linear function of the neutrino energy:
$$
\sigma_{\nu_\mu}^{CC} \approx 0.75 \times 10^{-38} (E/\textrm{GeV})\,\textrm{cm}^2
$$
and for muon-antineutrinos
$$
\sigma_{\bar{\nu}_\mu}^{CC}\approx 0.35 \times 10^{-38} (E/\textrm{GeV})\,\textrm{cm}^2
$$
I would like to know if there exists a similar approximation for the case of CC interaction for $\nu_{e}(\bar{\nu}_e)$.
I fail to find any similar question, but if there is any out there I would appreciate if you redirect me to it.
 A: In this energy range, the cross-sections for electron neutrinos scattering from atomic nuclei should be approximately the same as for muon neutrinos, since the W and Z bosons couple equally to all neutrino flavours.  The cross-sections only differ for neutrino energies close to the threshold for producing the charged leptons, as shown in Figure 10 of the paper you cite for $\nu_\tau$ charged current interactions. (Here is a non-paywalled version of the paper.) The energy range $1-20\,\textrm{GeV}$ is well above threshold for muon production, so $\sigma_{\nu_e}\approx\sigma_{\nu_\mu}$ and $\sigma_{\bar{\nu}_e}\approx\sigma_{\bar{\nu}_\mu}$. (Around $1$ GeV the muon mass still affects some higher order terms in the $\nu_\mu$/$\bar{\nu}_\mu$ cross-sections at the $\lesssim10\%$ level.)
The cross-sections do differ for scattering from electrons, since electron neutrinos and anti-neutrinos can scatter via charged-current processes not possible for muon and tau neutrinos, as shown in Figure 3 of your cited reference.
The cross-section for scattering from electrons is, however, negligible compared to scattering from nuclei in this energy range.
This is because although the neutrino-electron couplings are comparable to neutrino-quark couplings, the  cross-sections are tiny because the target electrons are so light. Neutrino cross-sections in this energy range are proportional to the centre-of-mass energy squared ($s=E_{CM}^2$), which for a stationary target (mass $m_{target}$) is
$$s=2 m_{target} E_\nu + m_{target}^2 + m_\nu^2 \approx 2 m_{target} E_\nu$$
So the cross-section is proportional to the target mass. Since electrons are 2000 times lighter than nucleons, the cross section for scattering from electrons is proportionally smaller and can be neglected if you are only interested in the inclusive total cross-section approximations you give for isoscalar targets. ("Isoscalar" means there are approximately equal numbers of protons and neutrons (and electrons).)
