My favourite bass I own is a MIJ fretless jazz. No problems at all with it, it is a well put together insturment. Even though it isn't a P bass I own, I think that the quality of the japanese fenders is pretty consistent. (unlike some of the MIMs)
Also I think there are some things about the appearance of MIJ's that make it unique too. For instance, the fretboard on mine appears to be just plain rosewood (no markers) with a black body and a w/b/w pickguard. I have never seen the particular fretboard I have on any fretless jazz bass other than another japanese. Mabey i'm wrong and they offer this neck on the others but i'm pretty sure it is exclusive to the japanese models, which I don't mind, because in my opinion it is the most beatiful fretless neck i've seen on a j bass. I'm not a big fan of the ones with fret markers that interfere with the beatiful look of the plain wood. It gives it a nice old style in my opinion. I would highly suggest a japanese made fender. It's worth it quality wise.
I'm attaching a picture of the only other one like mine i've seen. This one is exactly like mine but it has white strings and mine are black, but that is merely a cosmetic difference.