It literally means "10,000 years old", implying a very long life, but it's better to think of it just as a inspirational cry, like "Go for it" or "Hurrah", since, like most phrases, no one interprets it literally when they say or hear it. Americans tend to link to being specifically a battle cry (like in CoD) but I don't think it's seen that way to the Japanese themselves. Similar to how people think of "Kamikaze" meaning "suicide attack" but it just means "divine wind".
Banzai is "10,000 years old". It's an expression people in Sinosphere countries used to greet their kings/queens in ancient time, meant to wish them to have a long life. The translation here would be "Long lived Shinki-sama!"
I, I'm sorry, Yumeko-chan...Alice...Banzai!!Listen to me, Yumeko-san. That's not it!!Wait!Shinki-sama...So that is the way you have decided.
Fine.Yumeko-san!!Oh?
Yuki and Mai...
......... I see.PUNKEDSorry.Misunder... standing...
Eh? Did you say from Shinki-sama!?HakureiIt's a prank!Shinki-sama!!!
Didn't you pass away...!?Flower Shop
KazamiIt's useless trying to stop me now.
We've just gotten to the good part, so please don't get in the way.I heard about everything from mum... I mean, Shinki-sama. This is all a misunderstanding.