Hello
The local file won't work in a normal Silverlight application because of Silverlights "sanboxed" security model.
Running a normal Silverlight application in your browser, you can't access local resources like you can if you were running an installed winforms/WPF application.
Have a look at this video tutorial http://www.silverlight.net/learn/videos/all/local-file-access/, if you want to learn more about accessing local files using Silverlight.
I created a sample app and pointed it at your mp3 file and it' wouldn't work, however a quick search for sample mp3s, lead me to another freely available (first hit). Using the code/url below, it works ok.
myMediaelement.Source = new Uri("http://www.robtowns.com/music/blind_willie.mp3", UriKind.Absolute);
In regards to the loading, it will start downloading the file when it hits the code that sets the source of the mediaelement.
So if you set the source for the media element in your constructor or in your loaded event, it will automatically start downloading in the background.
Then you just call myMediaelement.Play(); on the button click event.
Hope this helpful