When it comes to taking long-haul flights in economy (which let's face it, almost 99.9% of us fly economy), there is no real way to be completely at ease BUT there is plenty you can do to make it as enjoyable as possible. The number one tip I have is to be prepared long in advance. There's nothing like stress to set off your trip on the wrong foot. I tend to pack a full few weeks in advance, and while I don't recommend this necessarily, it is good to have all your ducks in a row a few days before your flight.
Depending on where you're setting off from, you're most likely looking at an at least 13-hour flight to Japan. If you don't have the cash to splurge for first class - and let's face it basically no one does - see if your airline has to ability to upgrade to upgrade your seats for some extra legroom.