Start with the Apple/Universal preset then change:
Picture Settings>width: 480 (psp native) and anamorphic: none
Video> Average Bitrate:350 (low, but looks fine for a small screen and I like to have a small file to be able to fit more on my card)
Audio>AAC Stereo 48khz 128kbps
If your PSP says it can’t recognize the file, make sure you used the 32 bit version of handbrake.