You can pretend you are leaving and peek in through a window, or listen by the door. When he starts barking, do something to distract him like tapping on the window, or making some type of noise.
I live in an apartment also, so I needed to stop Felix's barking even though I could have just ignored it to go away. He is deaf though so they only thing I could think of is a spray bottle of water. I only use it if I know he doesn't need to go out (he barks in the morning when its time to go) I give him a spritz from around the corner so he doesn't know i'm there, well I hope he doesn't know
Usually it is enough to distract him and he forgets what he was barking about. He is getting much better, when he objects to the crate its only for a few whines then hes over it.
Do you ever leave him in the crate while you are home? I think letting Felix "practice" being in there while we are home helps too. They just don't want to be alone
Also, try mixing up your routine in the morning... walk out the door and come back in before he starts barking in and out- maybe he will realize that you ARE coming back sooner or later.
Also, do you leave him with some irresistable treat? I was feeding him before I left b/c my SO wakes up after me to take him outside again. but today I gave him a kong with frozen canned food.. Mmmm that must have been good!
I think it would be neighborly of you to thank the lady next door for letting you know about his barking, and let her know that you are doing everything you can to fix it. Maybe even ask your other neighbors if his barking his bothering them, and if so- tell them you are working on it too.