The rules vary by stadium
Unlike many Western stadiums with strict no-outside-food policies, Japanese venues are generally more relaxed. However, rules differ between baseball stadiums, soccer grounds, and basketball arenas.
Baseball stadiums
Plastic bottles: Usually OK
Most NPB stadiums allow sealed plastic bottles. Some require you to transfer drinks to paper cups at the gate.
Cans and glass bottles: Not allowed
For safety reasons, cans and glass are prohibited at most venues.
Food: Usually OK
Most stadiums allow outside food, including bento boxes from convenience stores. That said, stadium food is so good in Japan that you'll want to buy inside.
Soccer stadiums
J-League stadiums generally allow plastic bottles with caps. Alcohol brought from outside is usually prohibited — buy beer inside instead.
Basketball and volleyball arenas
Indoor arenas tend to have stricter rules. Outside food and drinks may be restricted. Check the venue's website before the game.