Yes. Its basically how my network is.
Modem --- WRT54Gs wireless for my dad. Ethernet to ---- WRT54Gs wireless for me
Flash your router like i had TGM do with hyperwrt or thibor so u get more control. Plug it into that patch cable. Open the router config, go to "Basic Setup".
Set the router to capture(setupcontent,dhcp). Name the router (ie apartment) and set MTU to capture(share,auto).
Scroll down to local ip. Make it something that will be different than the router already installed. my main router is .1 and my second is .100. Next is DHCP, disable it. Than go down and disable spanning tree.
Next go to basic settings for the wireless. Choose mode type WDS-AP. Next is network mode, choose capture(wlansetup,mixed). Choose a name for the network. Choose a channel. Now i have two wireless networks on one property (increased range basically) so i use 6 on one router and 11 on the other. You can basically use whatever you want in your case. I have SSID enabled and bridge disabled, you can really make these whatever you want.
With that setup you can have both routers running. I have ethernet and wireless on my dads, and just wireless on mine. Alot of those settings can be different, but it was hard to get them working in my case because of the two wireless signals, and this combo worked for me. I'll take some screenshots in a few for ya cuz i can be hard to follow sometimes.