Well it depends a lot, and a definite answer cannot be given unless you specify the player model. Sony has made several kinds of players, and while all those I've seen require Sony's program to transfer music, they might have other kinds of things as well.
In the easiest case you can just drag the files onto the player (like you can drag files onto a USB memory stick, for example). You can try it like this: plug the player in, open it in your file browser (Nautilus on Gnome, Konqueror on KDE, ...) and try copying some music files there the usual way (from menu, Edit -> Copy on the file, then Edit -> Paste in the file browser with the player open). Eject the player and see if you can play the file.
If that does not work, your next best try might be the usual audio apps on Linux desktops - Amarok on KDE, Rhythmbox on Ubuntu (or Banshee or whatever you have). Plug the player in and see if it appears somewhere in the player app; if it does, you can probably drag the audio directly from the application's music list onto the player list.
If neither of those works, chances are the player requires software that only Sony has officially made, and in that case you'll need to know the exact model number and start searching the web for a piece of software that is written specifially for that model (or player type more generally). If you post the player model number here, somebody might be able to help you out -- personally I don't own Sony's players, but I might be able to help searching for answers nevertheless.