Yes, a lens with a larger aperture (numerically smaller f/number) will produce a shallower depth-of-field, and a more blurred background. However, there's another factor working in the 5D's favor: It has a "full frame" sensor, the same size as 35mm film, while your camera has a smaller "APS-C" sensor. The larger sensor results in a shallower depth of field, for the same composition and aperture setting. See Matt Grum's comparison in this threadMatt Grum's comparison in this thread. So the 50/1.8 will be an improvement, but may not reach what you're seeing from your friend's 5D.
Also, you said you don't like zooming in with your 18-55 lens, because you prefer to work closer. Keep in mind that a 50mm lens will be similar to the long end of your zoom, so it may not mesh with your preferences.