I have a map depicting 500 points in a city. I need to create one layout per point, showing each point at the center of the layout, and all the other points that lie within a particular distance of the central point. The desired output format is a set of 500 distributable files that can be printed by an average computer user (pdf, png, jpg, etc...).
What's the best strategy for a project like this?
You should be looking at the Data Driven Pages functionality of ArcMap. At least, I assume you are using ArcMap and not some web app. It will allow you to automatically output all of your points as separate PDFs or some image format. Set the Data Driven Pages index to your point Feature Class (it is not required to be polygons, despite the examples in the page linked below), or alternatively a buffer around each point:
See:
What are Data Driven Pages?