Motivated by the recent synthesis of two-dimensional monolayer AlSb, we theoretically investigate its ground-state and electronic properties using the first-principles calculations coupled with Bethe-Salpeter equation. An excitonic instability is revealed as a result of larger exciton binding energy than the corresponding one-electron energy gap by $\ensuremath{\sim}0.1$ eV, which is indicative of a many-body ground state accompanied by spontaneous exciton generation. Spin-orbit coupling is proven to play a vital role in the prediction of the ground state. At room temperature, the two-dimensional monolayer AlSb is expected to transform into a direct gap semiconductor with phonon-limited electron and hole mobilities both around 1700 ${\mathrm{cm}}^{2}/\mathrm{V}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$. These results show that monolayer AlSb may provide a promising platform for realization of the excitonic insulator and for applications in the next-generation electronic devices.