Your two sentences cover the two cases of using negation to say the same thing.
In colloquial American English, you will encounter "She is not poorer than me" or "I am not richer than her." You will also encounter the more grammatically correct "She is not poorer than I" or "I am not richer than she." In these cases, speakers will often also add the correct form of the verb "to be" to the end of the sentence, since English speakers have become uncomfortable with pronouns at the end of a sentence, so "She is not poorer than I am" or "I am not richer than she is."
I bring up the issue of pronoun discomfort because American English speakers now overcorrect usage of "I" vs. "me" in many situations. In positions where "me" should be used, e.g., as the object of a preposition, English speakers will use "I" instead as an attempt to sound grammatically correct. This has become so pervasive that one can assume it will eventually "become" grammatical. See the Wikipedia article on hypercorrection for more details.
You will not be misunderstood if you use either pronoun variant in these example sentences.