Airport Express (Wireless N version) Problems

I have an airport express and it works fine when connected to the internet through a Cisco Linksys® wireless g router (pretty sure it's the WRT54GL, but I'm not at home right now so I can't check). However, when connecting the airport express directly to the Motorola Cable modem via Ethernet (out) it will not connect to the internet on my computer in the browser. The airport icon says it's connected though. I have to use the wireless routers wired ports in the back first to connect to the ethernet of the airport express. This was fine for a long time, but then the wireless g router started failing a couple days ago. I have to keep resetting it every ten minutes (unplugging and then replugging the power). Should I buy a new wireless g router or can I use the cable modem directly to the airport express (like it is supposed to be able to be used anyway)? Any insight would be appreciated, sincerely.