Since PSP have really shitty browser you can really download anything from ifile and that sucks. So, thanks to tomoya83, who have pushed me to this thought, I wrote a guide for you how to install and use Opera Mini on your PSP. And yes, opera mini is way better than PSP default browser.
Anyways, let's get started
You need Custom Firm Ware (CFW) (or google for singer and sign homebrew needed here)
PSPKVM (It's java emulator) Download
Any java browser for mobile phones. Google for them
1. Download PSPKVM, extract it from archive and copy folder named pspkvm to ms0:/psp/game
2. Download preferred java browser UCWEB
(was confirmed to work by zarcaraz)
3. Copy downloaded .jar of file of web browser to any where on you PSP ( I recommend root since it will be easier to find, though this is up to you)
4. Make sure you have wi-fi connection configured on your psp
5. Launch PSPKVM from game menu
6. In PSPVMK press "system midlets"
7. Press then "find applications"
8. Now navigate to jar file of your web browser
9. PSPKVM will now install your midlet and launch it (I hope you know that psp resoultions is 480*272 so select it when asked)
10. From now on it will be available under "Installed midlets"
Note: There is completely different way to type then in internal psp things. Keyboard is one of mobile phone
Square -> .,1 and other symbols
Up arrow -> abc2
Triangle -> def3
Left arrow -> ghi4
Right Trigger/Left Trigger + Circle -> jkl5
Right arrow -> mno6
Right Trigger/Left Trigger + Square -> pqrs7
Down arrow -> tuv8
Right Trigger/Left Trigger + Triangle -> wxyz9
X -> space,+0
Right Trigger/Left Trigger + X -> Backspace
Select -> Left functional button
Start -> Right functional button
Right Trigger/Left Trigger + Select -> *#
Right Trigger/Left Trigger + Start -> Capital/lower case letters
If you don't like this, you can:
a) Go blame Sony for never doing good PSP browser
b) Download from pc and then copy to PSP
If you still insist on downloading directly from PSP, good luck with this.