I have access to a wireless network at my flat, but require a wired connection for my IP-telephone to work. Is it possible to use a wireless router to "pick up" the signal and connect the IP-telephone to the router? If not, what kind of equipment would I need to do so?
Yep. Besides "client mode", also boxes called "workgroup bridge" (mostly 3com) and "game adapter" will work. Many of the "portable router" boxes also have a "client mode".
Personally, I would suggest getting a box that can do just about anything depending on settings. Linksys WRT54G with Sveasoft Alchemy or Talisman firmware are fairly universal and can do: wireless router or access point client mode (with multiple devices) WDS repeater mode AdHoc mode I don't think it will do transparent bridge (without WDS), but you probably don't need that.
Client mode with multiple MAC addresses is ideal for your setup. Use the client mode to connect to the apartment network. Plug in everything else into the LAN ports. Of course, this assumes that apartment network doesn't have some type of security system in place that will limit the number of devices you can connect.