In answer to the question of how to find info, go to the source and ask a cop/retired cop. Ensure the cop knows the location.
I is one-there are a few of us on AW. (My info applies only to a US setting and I don't know where your story's set.)
Re your scenario, is this an apartment building meaning a four-plex or six-plex, or do you mean a very large complex with hundreds of units?
For what crime is the bad guy wanted? Robbery, sexual assault? Or did he steal a car? The type of crime does influence how much effort will go into going after him.
Re a guy getting away by going into a building, and presumably he doesn't come out when the good guys knock on the door. Is it the bad guy's res? Gotta go get a search warrant. This means typing out an affidavit or giving sworn testimony before ye olde judge.
Re pursuing him and he goes into a building, is this hot pursuit, as in they are chasing him, in sight, a few yards behind, ready to grab 'im? In the US, this is one of the exceptions to needing a search warrant. Could be door-kicking time, tho I need more scenario info to say for sure.
Re how to get him out assuming the two cops are having to wait for a warrant or are attempting to negotiate him out or waiting for support...at any rate, waiting for whatever good reason. One cop is immediately going to the back door, backed off far enough to see window exits. As in, if it's a flat, say we know the apt covers the entire ground level, one cops positions to be able to eye the north and west sides, the other is opposite corner, able to see the south and east sides...because when we do go kick the door, he'd better be in there rather than us having screwed up and no watched all ingress and egress from the minute he went in his hidey-hole. This is something someone is going to swear to to get that warrant.
Re the possible hostage situation. Is this domestic? Stranger hostage? Was the guy armed? Threatening? You say it's only possible, so you must have more of the scenario in mind, but I dunno what you're imagining. Very good chance the commanding officer's next move (assuming no answer at the door and absent exigent circumstance that call for door kicking) will be to have dispatch rummage through the computers and come up with a phone number for the apt unit, call and tell the dude to get his ass outside. (I was a negotiator...I actually said it a lot nicer than this.)
Really, you can just show up at your nearest PD and ask. Approach any cop and ask. Say why you want the info and ask for some Q&A time.