Ok I'm guessing you're using wireless for internet as well ? You can't be connected simultaneously to two wireless networks, and your AirPort Express' streaming counts as a full-fledged network. And yes, unless your airPort is connected to the internet, you won't be able to get on the internet through it.
My linksys isn't wireless actually, I use my airport express for that.
What I believe you can do is configure your airport express to connect to your Linksys' wireless network, then you should be able to access it via your linksys' wireless connection. I don't have two WAPs so I can't test it myself, but it basically consists of setting up your airport express like this (screenshot) or something similar.
These two shots come from the AirPort Admin Utility. The important pull-down box (mode sans fil) should be set to "Connect to an existing wireless network", then enter your linksys' wireless network name and any relevalt security information, like WEP password, if necessary.