TL;DR: No, they do not serve sushi on planes. (See footnote.)
To set the stage, I've lived in Japan for several years, have traveled the length and breadth of the archipelago, and have flown in and out of the country on a wide selection of airlines (JL, NH, SQ, OZ, MH, AY, QF, TK, SK, NW, TG), in both economy and business class — and I've never seen actual raw-fish-and-rice sushi on a plane, unless I brought it myself.
First, on domestic flights in Japan, there is no food served at all in economy or business class, much less sushi. On JAL, a select few routes do offer simple meals in first class, but sushi is not on the menu. You're welcome to bring your own food on the plane though, including sushi, and even liquids are allowed on dom flights, although they may be scanned.
Second, on international flights to Japan, the food will be prepared not in Japan, but at the catering service(s) at the other end, which prepare food for all flights at all airlines, not just Japan. Are they going to offer sushi? No.
Third, on international economy flights from Japan, you've got Japanese catering firms that could offer sushi... but don't. Economy meals have to be cheap, reheatable and palatable to all, but as @jwenting states, sushi is expensive and perishable, and there are a lot of people -- yes, even people visiting Japan -- who dislike raw fish.
That leaves only one option: business and first class flights from Japan, where ingredient cost is not such an issue and there are usually multiple choices of appetizer. But sushi is still not a great option, because raw fish has to be kept cool, but the rice turns into a hard, gluey mess if goes cold, an impossible equation in the confines of an airplane. In business class, I have occasionally seen cooked types of sushi (grilled unagi eel, slices of futomaki rolls, etc), but I can't recall ever seeing raw fish on a plane.
But rejoice! If you've got more cash to throw around than I do, you can still enjoy raw fish on a plane... in First Class, since you simply can't do a proper Japanese kaiseki meal without sashimi. Here's the sashimi course from Asiana on a NRT-ICN flight:
(courtesy tng11 on Flyertalk)
And here's a trip report showcasing ANA's kyo-kaiseki course, also incorporating raw fish (well, slightly seared aburi, but close enough). JAL and Singapore Airlines offer kaiseki as well.
But note that none of these offer sushi (raw fish on rice), opting instead for sashimi (just plain raw fish). This is both more accurate, since you wouldn't serve sushi as part of kaiseki, and doesn't run into the aforementioned problem of trying to keep the fish cold and the rice warm simultaneously.
Footnote: I'm defining "sushi" as "raw fish on rice" here, which is narrower than the actual definition, but think of it this way: if someone offers you a sushi meal, and it has no raw fish, you'd most likely feel a bit cheated, no?