Been to Japan some 14-15 times now and have never had jet-lag going there and perhaps once or twice on the way home. At home I go to bed between 9-10 PM and wake up at 5 AM. I exercise quite a bit. I don't drink alcohol or much caffeine. That's for starters. On the plane, stay away from alcohol and caffeine. Take those catnaps when you can. I usually doze for 30-45 minutes a time 4-5 times a flight. Without fail I tend to fall asleep once the doors are closed and wake up just after takeoff. Wierd how it always happens.
When I arrive, if I am going to Tokyo I usually crash for 20-30 minutes on the Narita Express. Otherwise it's a nap on the longer bus ride to Tsukuba. When I get to the hotel I make sure to stay up until at least 8 PM. Since I'm used to waking up early I don't mind getting up at 4 AM the next day (or even 3 AM). Since I am active at home, if I bring my bike I go for a ride. Otherwise I'll run. Exercise helps fight jet lag as well. I always have meetings the first morning I'm there. I'm always perky, though after lunch gets challenging. But hey, even here the post lunch crash happens to me.
Going back is more of a challenge. Nothing is worse than a late afternoon flight from Tokyo that puts me home mid-afternoon. I just have to stay up a little longer. I take a 1 hour maximum nap in the afternoon, suffer, maybe go for a ride, and then go to bed between 8-9 PM. The next couple of mornings I typically wake up between 330-430 AM but I am fully rested.
So bottom line - catnaps, no alcohol, no caffeine, exercise.
Be sure to call the airline to confirm your vegetarian meal. I've had them forget mine a few times. What to do on the plane? Depends on the airline. When I flew United a few times I hated it. This was Houston-->San Francisco-->Tokyo. Service was not spectacular, movies were washed out, sound quality poor, and it was a common monitor. I much prefer the 777s of Continental and American with the individual entertainment systems.
For strict vegetarian I can't help out too much. I'm pesco-vegetarian and never have issues. There's always fish to eat. Otherwise, just about any restaurant will be fine. In my opinion the best tempura is vegetable tempura. The Japanese sweet potato should still be in season. I hate the sweet potatoes in the US but can't get enough of the ones there. Soba is vegetarian too (water, flour, soba sauce). One of my favorite restaurants in Tokyo is Maisui near Shimbashi Station. Great atmosphere and very good food. Best potato pancake I've ever eaten. Look for maisui and shinbashi on google and you'll find a link. If you get in a real pinch, you could always go to someplace like TGI Fridays (there's one across the street and up the hill from the Shinagawa Prince) for a salad. Just don't tell anyone you went to Japan to eat at Fridays. In reality, there are plenty of Japanese foods made without meat or fish.