Was your other thread not good enough?
@StriderNo9: Wait, you do know you can just connect a LAN port (not WAN) of any additional router to your first router and it'll basically just work, right? Pretty much all you have to do is set the second router's IP to anything but your first router's IP (most likely 192.168.0.1), so for example, 192.168.0.21 or something. You may have to set the second router's connection type to static IP and 192.168.0.1 and disable DHCP server, but it might not even be necessary. That's the beauty of ethernet, though--totally daisy-chainable.
I'm familiar with neither of these products, but here's the basics on what you're looking to do.
Your Arris router shouldn't have an internet/WAN ethernet port since it's also the modem. Connect the two switches/routers with a standard ethernet cable into the switch ports (not the Internet or WAN port). You don't need a X-over cable for this.
Because your Arris is the router and modem in one, I suggest using it as your DHCP server and gateway. You actually couldn't use the Apple one as a gateway anyway. Be sure to turn off DHCP and NAT on the Apple device. Turning off DHCP on the Apple device means you'll have to manually set the IP address on the Timecapsule. Make sure you assign it an address that's within the network for the Arris (192.168.1.x by default I'd imagine) but outside the assignable range for the DHCP server.
If you turn off the router functions on the Apple Timecapsule, you won't have to do change anything on the Arris for it. It will simply act as a switch.
If you want to segment your network into two different networks that connect at the modem you can do so by using NAT on both devices - I highly advise against this though. There's too much that can go wrong that could be as easily achieved with subnetting.