you can only download and play games on your psp that i know of by installing custom firmware on your psp.
and just incase your next question is ''can i get custom frimware(CFW) on my psp even tho its really high up in the orig.firmware?'' the answer is yes.
i did my psp from OFW 3.90
If you search the forums, there are lots of guides and tutorials that will show you what you need to do, I think you need a Pandoras Battery to go from 3.80 firmware though.
If you only have your psp to work with, then follow my guide on how to install CFW with only 1 psp in the tutorials section. It's the ONLY way to do what you want to do.
Founding father of the Wiibrew Packgen.
About 2 noob posts away from going ape sh*t