I'm thinking about getting a second hand PSP, the original one with UMD Drive. I just noticed that the PSN Store has many PSOne classics (like Final Fantasy VII and IX), but I wonder if that works on a PSP, or if it needs a Playstation 3, which I don't have?
Yes, all PSOne Classics from the PlayStation Network can be played on all the different versions of the PSP (PSP-1000, PSP-2000, PSP-3000, PSP GO and PSP-E1000).
Just make sure you have downloaded the latest firmware on your system.
PSOne Classics, launched in North America on December 4, 2006, are re-releases of original PlayStation games through emulation on the PlayStation Store.
Initially only available for download to the PS3 and playable on the PSP. System software version 1.70 for the PS3 allowed PSOne Classics to be playable on the PS3.
Eventually the PSP PlayStation Store was added in system software version 5.00, allowing the direct download of PSOne Classics to the PSP.