It there any known way of obtaining the Banach fixedpoint theorem from the Tarski fixedpoint theorem or viceversa?

Hello, I just found the question, so the answer might come a bit too lat, but.. Have a look at: Paweł Waszkiewicz, "Common patterns for metric and ordered fixed point theorems.", In Proceedings of the 7th Workshop on Fixed Points in Computer Science (Luigi Santocanale ed.), 2010, pp. 8387. I attended this talk last summer, and it addresses exactly your question. 


As suggested by Jacques, I turn my comment into an answer. This is not exactly what you ask for, but it is related. Efe Ok in Section 3.4 in Chapter 6 of his yettobewritten book on ordered sets gives a proof of the Banach fixed point theorem using the KantorovitchTarski fixed point theorem: 


Look the article : M.Jawahiri, D. Misane, M. Pouzet. Retracts: graphs and ordrerd sets from the metric point of view. Contemporary Mathematics, 1986, vol. 57 pp. 175226 

