The cable you linked to can connect an S-video output from your computer to an S-video input on your TV. But you don't have one! So no, this won't work. You need either:
(a) a different TV with a S-video input;
(b) a different video output on your computer, called "Composite Video", and a different cable with RCA connectors on all three lines; or,
(c) a converter to make Composite Video from an S-Video signal.
The better of the lot might be a different computer video card with Composite Video output available.
Here's the color coding convention on RCA cables:
Yellow is Composite Video
White is Audio Left (Stereo) or the only audio if it's mono
Red is Audio Right
In your case to make stereo audio output fit into a mono input on the old TV, you can do:
(a) White to White - your TV will get only the Left channel of the Stereo signals, but you can ignore that.
(b) White to White and then, in the computer's audio card properties, find a control which converts any stereo signal into mono output on the Left (White) connector only. Then your TV will get both channels mixed together on one mono line.
(c) Get a 2-to-1 RCA splitter with two female connectors for the white and red lines coming from the computer, and a single male output connector to plug into the TV. This will effectively connect the Left and Right audio signals together in parallel and feed them both into the TV.