"I shall be rich" suggests intention (to become rich somehow), whereas "I'm going to be rich" is more definitive, suggesting it's inevitable, like you have a pretty surefire plan or prior knowledge of inherentence or a windfall. Shall is very much in use.
The idea is that "shall" is strictly future whereas this sentence gives us a grammar exercise using the "near future". So while you may naturally use "shall" to refer to future events, it's necessary to recognize the slightly different nuance of the "near future" in this case.
That means "I want to be rich" which is different to "I am going to be rich"
I understand it's probably not a literal translation, but would you be able to translate it with" I will become rich"?
Not a native speaker, but I believe that would be closer to "Je vais devenir riche". Devenir = Become