OP, are you able to sleep through the construction now? Have you tried napping during it? That is the big issue to solve at the moment - find something that works for you (headphones, white noise, earplugs, something). Truly, if nothing works, you may need to look into ways to better soundproof your apartment.
The baby will be able to sleep through it. The PP was right that the baby gets used to sounds they hear in utero. For instance, we have a dog with the loudest howl you've ever heard, and she goes off whenever anyone comes to the door. We were terrified she'd wake the baby. Guess what? Baby didn't stir one bit when the dog howled. He reacted to lots of other random noises, but not that.