When Momaday mentions that his grandmother, Aho, belonged to "the last culture to evolve in North America," he is referring to her heritage in the Kiowa tribe of Native Americans. There are many quotations that show Aho's devotion both to the Kiowa heritage and the Kiowa religion.
[Aho] belonged to the last culture to evolve in North America. Her forebears came down from the high country in Western Montana nearly three centuries ago. ... Although my grandmother lived out her long life in the shadow of Rainy Mountain, the immense landscape of the continental interior lay like memory in her blood.
The sentence immediately following the quotation you give in your question gives us, as readers, a better indication of Momaday's grandmother's heritage. The Kiowa heritage, then, is almost three centuries old and originates from the western part of Montana. In asking the question you have asked, it is clear that your instructor wants you to realize that Aho is not only Momaday's grandmother but also a member of the Kiowa tribe of Native Americans. Aho lived "in the shadow" of the Kiowa landmark called Rainy Mountain. Further, Aho's body is also buried there. We learn soon that is the reason why Momaday travels there, to visit her grave.
Now that I can have her only in memory, I see my grandmother in the several postures that were peculiar to her: standing at the wood stove on a winter morning . . . sitting at the south window, bent above her beadwork . . . going out upon a cane, very slowly as she did when the weight of age came upon her; praying. I remember her most often at prayer.
This "last culture" that you mention in the quotation also has a lot to do with the Kiowa religion. In the quote above you can see the mention of many aspects of cultural heritage: wood stoves, bead crafts, etc. The reader can also see that prayer was important to Aho. As a devout member of the Kiowa tribe of Native Americans, Aho would be praying to the Sun Dance god called Tai-me. Aho developed this devotion when she was just a child witnessing the famous Kiowa Sun Dances.