You don't get to decide how the user's fonts look. The exact choice of font and whether it is anti-aliased is a client-side setting. Some (terrible luddites, IMO) prefer the pixellated look. In any case, Linux is likely to be using a lookalike fallback font (eg. ‘Sans’) rather than Arial itself, so it's never going to look exactly the same.
To turn anti-aliasing on in Windows for most fonts (in all applications, not just web browsers), you need to enable ClearType. Without ClearType, Windows's anti-aliasing is the old broken ‘font smoothing’ option, which for most fonts is so poor that Windows prefers not to use it. ClearType rendering isn't quite the same as the anti-aliasing Linux will be using; usually it only anti-aliases in the horizontal direction. So again it won't quite look the same, but it's generally considered pretty good at least for the Latin alphabet.
If you're really desperate to push it from the server side, you could use a @font-face
rule to embed a font that has the GASP table hacked to always use anti-aliasing (using a ttfgasp.exe or a font editor). However doing this to Arial (or indeed most fonts without an open licence) would not be technically legal.