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 $sim$0.1 eV, which is an indicative of a many-body ground state accompanied by spontaneous exciton generation. Including the spin-orbit coupling is proven to be a must to correctly predict the ground state. At room temperature, the two-dimensional monolayer AlSb is a direct gap semiconductor with phonon-limited electron and hole mobilities both around 1700 cm$^2$/V$cdot$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.
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