Wie verbinde ich Router mit diesem alten Telefonschalter?

That should very much depend on your needs and the cables run in the wall, they could be at different speeds. Just changing the ports should definitely work, but maybe not optimally

Can’t tell for sure from this photo because some people modified their socket, and this one looks like it could have been modified. You need to know that you need a DSL modem / router to work with that (if it works at all). If you have a cable modem or something else then it will not work. You could check at the Deutsche Telekom website if your address offers DSL. Or ask the neighbours. If there is DSL in the area, then probably you could plug the DSL router into one of the three sockets. Normally, if the socket is not modified, you would plug it into the middle socket. But as this one looks like potentially they modified it, it could also be that you could plug it into one of the outer sockets and it would work. You just need to check if the plug fits – if it fits (it should go in easily, don’t use much force), then the DSL modem should be plugged in correctly and it should be able to connect to the DSL client on the Telekom side.