How to connect to modem setup as bridge through Airport Extreme?
The problem isn't just that you have two devices using the same 192.168.1.1 IP address, but also that two network segments are using the same subnet (the first 3 parts of the IP, 192.168.1.x). If you're on the AirPort's LAN at 192.168.1.123 and you try to talk to the modem at 192.168.1.1, the network sees "192.168.1" and recognizes it as local traffic, meaning that it never goes out the WAN port where the modem is.

It's sort of like if I try to mail a letter to "123 Fake St, London". I mean the one in England, but there's a city called London right here in Ontario, so that's where the post office is going to send it.

Anyway, when you made your own subnetwork with NAT, you should have picked a new IP subnet to go with it. The AirPort utility calls this part of the IPv4 DHCP range, even though it affects more than DHCP. It's probably currently set to "192.168.1.100 to 200". Just change the third part to something else, like "192.168.0.100 to 200". This will move your entire LAN to the 192.168.0.x subnet, and the AirPort will take 192.168.0.1 for itself.