Your computer should give you information about the IP of the computer that gave you that IP address. I know in XP it shows up in ipconfig's output.

Then ping that IP address. It probably makes no difference if it responds or not. Then look at the computer's ARP table (under Windows, "arp -a") and match that IP address to a MAC address.

Once you have the MAC address, you should be able to find what switch port the device is connected to, assuming they're not dumb switches. If they are dumb switches, you can at least use the MAC address to find the manufacturer of the NIC, which is often the manufacturer of the whole device.

To prevent it, you'd want to assign a different network to each apartment, and have them on different network segments, whether that's a different switch for each apartment, or, if you have decent switches, assigning a VLAN to each apartment. That way, they can only take themselves out. (You could also filter DHCP responses at the switch, but if your switch can do that, it can do VLANs, which is probably the better option anyway.)

If you don't want to do that, though, the only thing I can think of is to set up something that would try to suck up all of the offending device's DHCP leases, and that is going to be tough to implement.