How to configure your Router depends on the model.
If you connect to a "regular" LAN-Port in your router-device and turn on your rourter's DHCP there will be two DHCP Networks in the LAN, that won't work (regardless of the IP-settings)- period. Then you have to use the "Access-Point-Only-Function" of your router (no DHCP, no NAT, etc.). All the devices will get an IP (including Gateway and DNS) from your landlords DHCP. (You might want to talk to your landlord that he reserves an IP just for you and takes it out of the DHCP pool).
If you have a "special" port for an "incoming network only". You can turn on your routers DHCP and configure the whole thing with NAT enabled. Make sure you're router receives it's settings form your landlords DHCP.
Depending on your model/software your router-device might not even need an IP to be configured when it is in "Access-Point-Only-Mode". But if it does need an IP make sure to give it an IP that is not used within the network (192.168.0.X again, as long as it is not used in the network). You can also use an IP outside of 192.168.0.X e.g. 192.168.55.1 but then you need a divice with 192.168.55.X (anything but 1) to configure the router-device in the future.
I didn't use the official terminology, since you don't seem to be very familiar with it, let me know if you have any further questions.
PS. your setup is very similar to this one: WiFi cannot connect to internet in two router setup and this one: How to connect a wifi router to a cable modem router?
PPS. The same SSID is only required when you use multiple APs (one in your router and an additional one) but since you use your router as an access point this does not apply (unless you change your mind and want to extend your landlords Wifi so both of you can use the same configuration on your wireless devices)