Hi and welcome to FixYa,
Offhand, the only connection feature of the Sharp Sidekick LX that could be used is Bluetooth assuming that the new Sony Ericsson phone (what model?) has likewise Bluetooth connectivity. But this might be too slow for large files or large number of files.
Alternately, if the new Sony Ericsson phone has both an internal memory (like the Sidekick LX has 64Mb) and provision for external memory using the same type of SD card, then it would be possible to partially transfer/copy music files, a few at a time. The flow would be, LX's SD memory to internal memory of Sony, replace memory card, internal memory of Sony to external. To be repeated until all wanted files copied.
Of course, the fastest would be to go to an Internet Cafe and use a computer with a memory card reader.
Good luck and Thank you for using FixYa.