In this paper, we make a detailed discussion on the $eta$-meson leading-twist light-cone distribution amplitude (LCDA) $phi_{2;eta}(u,mu)$ by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments $langlexi^n_{2;eta}rangle|_{mu_0} $ with $n = (2, 4, 6)$ can be determined, where the initial scale $mu_0$ is set as the usual choice of $1$ GeV. Numerically, we obtain $langlexi_{2;eta}^2rangle|_{mu_0} =0.204_{-0.012}^{+0.008}$, $langlexi_{2;eta}^4 rangle|_{mu_0} =0.092_{ - 0.007}^{ + 0.006}$, and $langlexi_{2;eta}^6 rangle|_{mu_0} =0.054_{-0.005}^{+0.004}$. Next, we calculate the $D_stoeta$ transition form factor (TFF) $f_{+}(q^2)$ within the QCD light-cone sum rules approach up to NLO level. Its value at the large recoil region is $f_{+}(0) = 0.484_{-0.036}^{+0.039}$. After extrapolating the TFF to the allowable physical region, we then obtain the total decay widthes and the branching fractions of the semi-leptonic decay $D_s^+toetaell^+ u_ell$, i.e. $Gamma(D_s^+ toeta e^+ u_e)=(31.197_{-4.323}^{+5.456})times 10^{-15}~{rm GeV}$, ${cal B}(D_s^+ toeta e^+ u_e)=2.389_{-0.331}^{+0.418}$ for $D_s^+ toeta e^+ u_e$ channel, and $Gamma(D_s^+ toetamu^+ u_mu)=(30.849_{-4.273}^{+5.397})times 10^{-15}~{rm GeV}$, ${cal B}(D_s^+ toetamu^+ u_mu)=2.362_{-0.327}^{+0.413}$ for $D_s^+ toetamu^+ u_mu$ channel respectively. Those values show good agreement with the recent BES-III measurements.
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