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.
Pro tip
If in doubt, bring an empty water bottle and fill it at water fountains inside. Convenience stores near every stadium sell snacks in allowed containers.
Summary
Japanese stadiums are generally more lenient than Western venues about outside food. When in doubt, check the venue's official website or ask at the gate.