Where can I find an analysis of Oliver Optic's Rich and Humble? (1863)
While there does not appear to be any substantial analysis online about Rich and Humble: Or The Mission of Bertha Grant by William Taylor Adams writing as Oliver Optic, at least one book has written about the novel and analyzed it a bit.
One such book is In the Company of Books: Literature and Its Classes in Nineteenth-Century America by Sarah Wadsworth. Wadsworth discusses the novel in the context of the foreword written by Adams, saying the most interesting thing about the book is "the explicit claim it makes to target a female audience."
Wadsworth goes on to say that the novel does fulfill that description: it charts Bertha Grant's journey from luxury to poverty and back to luxury. However, she points out that James had written primarily for boys before Rich and Humble. There is also evidence in the text showing he wanted boys to have an entry point into the book—going so far as to address them in the foreword as well. She also points out that he includes multiple male characters and sporting events to maintain male interest.
There are several ways to approach analyzing the text when no previous analyses are available. One is to read it from a historical standpoint. The novel is set in New York. Learning about the sociopolitical climate of the time and relating it to the actions of the characters or the themes the author seems to be championing is one way to analyze it. You can also think of it with regards to the life of the author, examine the motives of a single character, or discuss one theme in the book and follow it from beginning to end. Just make sure to cite evidence from the text to support any claims you make.