The good news
Japanese stadiums are generally MORE lenient about outside food and drinks than stadiums in most other countries. Many baseball stadiums allow you to bring in your own bento boxes and non-alcoholic drinks. This is a cultural tradition — fans often stop at convenience stores or department store food halls to buy elaborate bento before the game.
Baseball stadiums
Most NPB stadiums allow outside food. Drinks rules vary: some allow PET bottles (plastic), others require you to pour cans into paper cups at the gate. Glass bottles are universally banned. Alcoholic beverages from outside are usually prohibited. Check each stadium's website, but the general rule is: food OK, plastic bottles OK, cans maybe, glass no, outside alcohol no.
Soccer stadiums
J-League stadiums generally allow outside food and PET bottles. Cans and glass bottles must be left at the gate or poured into cups. Some stadiums have stricter rules for specific matches (derbies, playoff games). Umbrellas are discouraged — bring a rain poncho instead.
Basketball arenas
B.League arenas are generally stricter about outside food and drinks. Most arenas prefer you buy from vendors inside. Small snacks are usually fine, but full meals from outside may not be allowed. Check the specific arena's rules.
Universal rules
Banned everywhere: glass bottles, fireworks, musical instruments (unless in designated cheering section), large banners without permission, selfie sticks in crowded areas, drones. Cameras: generally OK for personal use, but professional-looking equipment (long telephoto lenses) may require media accreditation.
Pro tip
Even if you can bring food in, try the stadium food — it's genuinely excellent in Japan. Each stadium serves local specialties that you can't get anywhere else.