The Apartment is first mentioned in Entry #43 by Alex as being the place he woke up in after trying to escape Amy and Jessica's House with Amy, having no memory of anything between then. Entry #44 is where we finally see his apartment as he wanders about doing odd thing. The Operator also makes an appearance, causing Alex to vanish at one point, but since he is in his apartment later it is assumed he either doesn't remember or simply ignored it. Entry #45 also pertains to his apartment, this time being that Masky and Hoody know where he lives and lure him out of his apartment, trying to kill him before running away.
Entry #46 has Jay breaking into Alex's apartment in order to search for clues. The Operator once again shows up, but this time Alex does not run, although Jay bolts and grabs Alex's spare front door key. Alex never states why he didn't run or why he didn't respond to the Operator right behind him. In Entry #50, Jay breaks into Alex's apartment again, this time during the day and steals a tape.
After the events of Entry #52, it is believed that Alex no longer lives here, considering the events that transpired.
The location itself doesn't seem very important, however the Operator appears there very frequently due to Alex's involvement with him.