I would suggest you talk to the landlord. you don't want to screw up something that the landlord had planned like having someone come in and terminate the connections. At the very least the telephone company would do that for their equipment if you order a land line.
Otherwise...if just on your own...
I would suggest you identify which is which in that cabinet. You appear to have a larger cable which I am assuming is the telephone bundle. Then you have the other stripped cables [I have only seen solid colors used for network cabling] which we assume is the network cabling. The dual outlets are top network and bottom phone.
I HIGHLY suggest you get a trained person to come in and terminate and setup for connection [they won't hook up the router or phone system] with labels so you can plug the correct cable to the correct connection.
Given the setup you should not have the network cables terminated with end at the cabinet but to a patch panel. This allows you to set what ports are connected to the router/internet and leave unused ports unconnected. You have many options. Telephone lines also can go into a patch panel allowing multiple phone numbers to each port to be assigned.