Phantasy Star Portable is part of the PSU series, not PSO. I've already beaten the demo, twice, despite it being entirely in Japanese. Not that difficult considering the menu are identical to those in the other PSU games. The actual PSP storyline bridges the storyline gap found between PSU and PSU: AotI.
Also, PSP does not have online play! SoJ is discussing the potential release of a downloadable patch that would add online play capability, but it is not yet an available feature.
Phantasy Star 0 (that's a zero folks) for the DS on the other hand shares many similarities with PSO and unconfirmed word on it is that Sega may end up announcing that the zero represents it as being the prequel to the PSO series, but that is just a rumor at this point in time. In all seriousness though, the developement team behind PS0 was headed up by former Sonic Team members who were responsible for DC PSO. All of the character classes of PSO are available in PSO, including the never before available HUnewm (Hunter Newman Male). And <drum roll>....
PS0 features online play!