The biggest problem is getting devices to talk to each other. In the case of an Xbox 360 and the Sony PSP, both devices will totally fail to recognize each other even though they both support the same hardware interface: USB. Another problem that sometimes crops up is that of file format. While both devices support the common MP3 format, neither will read the others proprietary formats (Windows Media Audio and ATRAC3, respectively.)
If the music files in question are in MP3 format, it's possible to read the PSP's music files using a Memory Stick--the PSP's storage format--to USB adapter, and then plug the adapter into the Xbox 360. As long as the memory stick is formatted in a file system that both devices understand, the Xbox 360 can read the files directly from the memory stick.
If the files are not in a common format, a computer will be needed to facilitate the transfer. If the files are in ATRAC3 format, you will have to use a program that can transcode ATRAC3 into MP3 (or you can use Windows Media Audio, but this will increase complexity.) Once the files are converted, it's probably best to put them on a standard USB flash drive for transfer to the Xbox 360.