CityStrides gets its data from OpenStreetMap. Mapbox is the software that displays the visual map interface. The list of countries/regions/cities/streets throughout the site is the data that I collect/compile from OSM.
Mapbox also gets its data from OSM, but may have some form of its own internal data structure/source as well.
Cities are on an automatic syncing schedule that attempts to sync batches every minute, assuming the current system load is not too high. Each city displays its last sun date on its page, just under its name.
While currently the rate seems to be every 3 weeks or so, this is not a reliable number. There is no set frequency, since the system is constantly operating and being affected by many things (size of updates, new cities being added, people running in new cities, etc).