AE Wifi works (No modem that I know of. How can this be?)
The AirPort Extreme is connected to another "main" modem/router that you cannot see because it is located back at the office, likely in the equipment room or cabinet. The AirPort receives its Internet connection information from the "main" router.
As long as the AirPort Extreme has an Internet connection.....it does......it is capable of providing a WiFi network to which other devices can connect. The "main" router assigns an IP address to each Ethernet or WiFi device that connects and the AirPort Extreme......which is configured to operate in Bridge Mode......simply passes that information through to connected devices.
Wifi does not work.
Did you power off the AirPort Extreme, leave it off for a few minutes, then power it back up?
If yes, the "main" router has likely been programmed to only allow a few devices to connect to the Ethernet port in each apartment or office. This is a very common way of preserving bandwidth and not letting one user "hog" the Internet connection.
If that is the case, then you can try to set up the AirPort Extreme as a router.......not a bridge like it is now.....and let the AirPort Extreme act as a router to assign IP addresses to devices on your network. This way, the AirPort Extreme will only require one connection from the "main" router and it will it turn assign different IP addresses to each of your connected devices.
If you decide to try this, you will need to connect the AirPort Extreme directly to the Ethernet wall port. If the AirPort Extreme does not have enough Ethernet ports for your needs, you can connect the Ethernet switch to one of the LAN ports on the AirPort Extreme.
There are both plusses and minuses for this type of setup. The advantage will be that the AirPort Extreme can assign IP addresses for up to 50 WiFi devices and 100+ Ethernet devices. Because the NAT firewall be enabled on the AirPort Extreme, you will have an additional layer of security on your network.
The downside is that type of setup will result in a Double NAT message. Double NAT simply means that there are two routers on the network instead of the normal one. This can sometimes result in connection slow downs and browsing issues. For example, you might be able to access some Internet sites but not others.
On a simple network, Double NAT does not usually cause problems, but there are no guarantees. AirPort Utility on your Mac will display the Double NAT status message, but there is a way to tell the utility to "ignore" the issue if all seems to be OK, so the AirPort will display a green light instead of blinking amber.
I use this type of setup when I am traveling with an AirPort router and the hotel has an Ethernet port in the room. This way, I can set up my own "private" network in the room, so other devices on the hotel network will not be able to see my devices and vice versa.
We can tell you how try the the Double NAT setup if you want, post back if you need more details.
Do I need to buy a modem/router and replace the Airport Extreme to get my first setup to work?
At the present time, you don't need a modem at all, since the Internet connection is being handled by the "main" router at the facility, which is likely a modem/router or gateway type of device that combines the functions of a separate modem and router in one package. Since you don't need another modem, you could add another router only .....not a modem/router.....instead of the AirPort, but you will still have the Double NAT issue to deal with.
If the Double NAT setup will not work for you, then your only option will likely be to contact an Internet Service Provider (ISP) to set up your own service in your apartment or office. Then, you will need a modem to connect to the AirPort Extreme. Or, if the ISP requires you to use their modem/router or gateway, then you really won't need the AirPort Extreme at all.
