High-level ab initio calculations on the ground and the excited states of aluminum monobromide (AlBr) have been carried out by utilizing the internally contracted multireference configuration interaction method plus Davidson correction (icMRCI+Q) method. The core-valence correlation (CV) correction and spin-orbit coupling (SOC) effect have been investigated in the calculations. The potential energy curves (PECs) of the 13 Lambda-S states, as well as those of the 24 Omega states generated from the Lambda-S states under the SOC effect, have been obtained. The spectroscopic constants of the bound states have been determined, which are in accordance with the available experiment results. The SOC induced predissociation mechanisms of the a(3)Pi and A(1)Pi states have been analyzed with the aid of the spin-orbit matrix element. The transition properties of 0(+)(2)-X0(+), 1(1)-X0(+) and 1(2)-X0(+) transitions are predicted, including the transition dipole moments (TDMs), Franck-Condon factors (FCFs), and the radiative lifetimes. Finally, the possibility of AlBr to be used for molecular laser cooling has been discussed based on our calculations.