First if you have the you tube downloader you can CONVERT INTO A PSP FILE THAn you can put them in your PSP without problem SO.1 STEP.YOU SHOULD HAVE THE YOU TUBE DOWNLOADER TO DOWNLOAD VIDEOS FROM YOUTUBE
2STEP. YOU CAN CONVERT THEM FROM YOU TUBE DOWNLOADER and you can delet than after you put them on your PSP . HOPE I HELPED

1. Download your youtube movies with FREE FLV converter (search on google for this).
2. Convert your files to Sony PSP(TM) with TOTAL VIDEO CONVERTER.

Maybe you have other software, but i'm using this one and it works great.

Upgrading flash player? Simply just visit Adobe website and download!

The two other answerers seemed to have missed one simple point, HE/SHE DOES NOT WANT TO DOWNLOAD THE VIDEO. Back to the question, The only way to upgrade the flash player on the psp would be to install custom firmware, which may involve some hacking of the psp.

This article may help you~
May it help~

you have to mod or have psp using flash enabled browser..

or

this will help you… use 'zillatube'

It will help download youtube videos and convert them to mp4 format fo PSP

This is the fastest way.

It works very well….just google search for 'zillatube'=
================================♥♥♥