Only a limited number of people were able to read and write, so the invention of writing probably didn't affect average people (farmers, craftsmen, etc) that much directly, but it is very difficult for us to determine (we don't have any sources for their world view).

There is of course evidence for cultural contact between ancient Mesopotamia and the surrounding regions (Egypt, Anatolia, the Iranian plateau, the area around the Persian Gulf and the Indus Valley). But I'm not sure to what degree we can assume that average people were aware of this - traders and members of the ruling class probably were to some degree and the invention of writing must have made a difference to them. The earliest preserved letters also include letters relating to a diplomatic exchange, although letters as a genre appear relatively late (in the 24th century BC; there are also interregional treaties from that period).

Foreign places are mentioned in some literary and mythological texts, but they seem to be rather 'mythical' and we of course can't be sure that they weren't already there in some earlier oral tradition.

There is quite an interesting letter from the Amarna correspondence in that regard (this dates to the late Bronze Age, so more than a thousand years later than you are asking for): It is a letter from the Babylonian king to the pharaoh of Egypt where the Babylonian king complains that the pharaoh did not visit him when he was ill.

"Furthermore, since I was not well and my brother showed me no concern, I for my part became angry with my brother, saying, "Has my brother not heard that I am ill? Why has he shown me no concern? Why has he sent no messenger here and visited me?" My brother's messenger addressed me, saying, "(It) is not a place close by so your brother can hear (about you) and send you greetings. The country is far away. Who is going to tell your brother so he can immediately send you greetings? Would your brother hear that you are ill and still not send you his messenger?" I for my part addressed him as follows, saying, "For my brother, a Great King, is there really a faraway country and a close-by one?" He for his part addressed me as follows, saying, "Ask your own messenger whether the country is far away and as a result your brother did not hear (about you) and did not send (anyone) to greet you. Now, since I asked my own messenger and he said to me that the journey is far, I was not angry (any longer), I said no more." (EA 7, cited after Moran, The Amarna Letters)

Of course some of this is diplomatic banter, but it is interesting to note that while the king had an idea that there was this ruler of another country, he had no clear idea of the distances involved.