O. Neugebauer, The Exact Sciences in Antiquity, Dover, 1969.
Chapter I, Numbers
Gives an introduction to ancient number systems, as well as some material on calendars. Pay attention to the parts on Babylonian numbers.
Chapter II, Babylonian mathematics
Read briefly to section 19 on p. 34. You can skip the more detailed mathematical matters, unless you have a strong interest in the subject.
Chapter III, The sources: their decipherment and evaluation
You can skip the more mathematical parts if you want. Just try to get a general overview.
Chapter V, Babylonian astronomy
Try to read the whole chapter, although I realize you may have some trouble with the more technical parts. Concentrate on the part up to p. 110. We will do Babylonian astronomy before Egyptian.
Chapter IV, Egyptian mathematics and astronomy
Read this.
Chapter VI
Skip. Greek astronomy was a big advance and is interesting, but is not related to archaeology.