Both are correct (αυτό after or before the noun, before is more common, but both are ok). The sentence sounds more native like this, though: "Εγώ το ξέρω αυτό το μέρος." Το could be omitted if the sentence continued somehow: " (Το) Ξέρω το μέρος αυτό, αλλά δεν έχω πάει ποτέ"="i know this place, but i have never gone there"
What exactly does one mean by "μέρος" ? Does it mean a limited space like a restaurant or does it imply a limited area with a number of streets, e.g. In New York, I know Greenwich village, I know that area of New York. Do you know " The Dirty Pig" Oh yes I know that place. So how do you translate the word area in this context? Thanks