On a whim, I thought I'd put this table together, chiefly for Chgowiz's benefit. I had done the research already, I just had to crunch it a little to find those cities that would fit the criteria.
The table below describes a number of cities founded before 1500 B.C., ranked from oldest to newest, which are still occupied today. The list isn't meant to be exhaustive or even particularly accurate - the figures merely correspond to research I've done. When it comes to dates more than 2,000 years old, there are always disagreements.
Nevertheless, if you were looking for a list of very ancient cities which might give you some ideas about the time period pre and post Mesopotamia, you could do worse. Disappointingly, the list doesn't include cities that once existed and are now long gone - which explains why there are few Mesopotamian cities listed. Most of the ones that exist today did not exist then. However, I'm sure Chgowiz already has a list of Mesopotamian cities. This might be a handy appendix to those.