Yes, PS3s are all backwards compatible with PS1 games, and some with PS2 games, though usually only older PS3 models.
Just a heads up though that the backwards compatibility can be funky. I've had a few instances where my PS3 can't run some PS2 games without glitching, or a loss in framerate.
Persona 3 can be purchased from the PSN for like 5-10 bucks if I recall, that's how I played it.
Though keep in mind even older games from the PSN store can have problems. I got DDS1+2 off of the PSN for my PS3 and while the game mostly functions well, audio likes to desync in cutscenes. I don't remember any issues when I played P3 though.