Omega!Alec would be similar to trans man!Alec in the sense of feeling even more intensely than in canon a deprivation, a sense that his body could be more powerful and more suited to his drives. He would be sexually frustrated at all times, and this would come forward in how sadistic he would be with his partners. He would love torturing alphas and keeping them enslaved to him through drugs, blackmail, manipulation, love, or more literal imprisonment. He would also have himself sterilized as soon as possible, probably in his mid to late twenties, to prevent pregnancy.

So, as always, evil man. He'd just have a more specific set of targets, although he'd torment his share of fellow omegas and betas as well.