Hi,
I hate to think I've got to grow up, and be Miss March, and wear long gowns, and look as prim as a China Aster!
'Little Women' is a classic story of young sisters whose family name is March. When Jo March says this, she means that she does not want to grow up and have everyone call her by her family name instead of just 'Jo'.
A China Aster is a plant. I guess Jo sees it as a symbol of sedateness. She may also mean, in irony, that a plant is not active and boisterous like she is.
Best wishes, Clive