The golden rule
Japanese sports culture values collective harmony. Your behavior affects everyone around you. The goal is to enjoy the game while being considerate of other fans. Follow this principle and you'll be fine in any stadium.
Seating sections matter
This is the most important rule. Home team fans and visiting team fans sit in strictly separated sections. NEVER sit in the wrong section wearing the opposing team's colors. If you're neutral, sit in the infield/reserved sections. Check your ticket carefully for section information.
Cleaning up
Japanese fans clean up after themselves — and you should too. Keep your trash organized during the game and dispose of it properly when leaving. Many fans bring small garbage bags. The stadiums' legendary cleanliness is maintained by the fans themselves, not just cleaning staff.
Cheering etiquette
In the organized cheering section (usually outfield in baseball, behind the goal in soccer): Stand, sing, clap, and follow the lead of the cheer leaders. In reserved/infield sections: Sit down, clap when exciting things happen, but don't stand for extended periods (you'll block the view of people behind you).
Don't do these things
Don't use an umbrella in the stands (use a poncho). Don't throw anything onto the field. Don't boo or jeer at the opposing team (respectful silence is the Japanese way to express disapproval). Don't get visibly drunk and disruptive. Don't stand in walkways during play.