If the light does not come on at all when connected, the shuffle is not getting power. It may not be a battery issue, since the shuffle should be getting power, even if the battery has become faulty.
Try pushing the plug on the USB docking cable completely into the headphones jack as shown in this document
Then, pull out the plug very slightly to see if you can get the light to come on.
There are three possible reasons for the shuffle not getting power. (1) A problem with computer's USB port not working, or not supplying sufficient power. (2) A faulty USB docking cable. Or (3) a hardware problem on the shuffle (such as a bad headphones jack or logic board).
To rule out (1), shut down (power off) your computer and remove all USB devices including hub (except USB keyboard/mouse if used). Start up computer, run iTunes, and connect shuffle to a direct USB port on computer. If you still get no light, connect something else to the same USB port (preferably something similar like a USB flash drive or port-powered external drive) to confirm the port is working properly.
To rule out (2), you'll need to try a different USB docking cable with your shuffle, or try your cable with another working shuffle that is 4th (or 3rd) gen.
If you can rule out (1) and (2), the cause is likely to be (3). If it's less than one year from purchase date ("not even a year"), it is under warranty (for manufacturing defects). If there is an Apple Store nearby, you can make a reservation at the store's Genius Bar, and take it there (take along your docking cable).