If you use third-party systems to update products on Prestashop (e.g. integrations with CRM  systems, CSV / XML dropshipping import modules etc.), the first thing to check is whether or not they use the standard methods via Prestashop classes to update products or whether they act directly with database queries.

This is a technical question to check with the developers of the third party system you are using.

In the first case (update via Prestashop classes), Fastbay will trigger itself during the product update by syncing the product in real time.
This may slow down the import operations due to the need to communicate with eBay, but it will ensure the real-time sync of your catalog.
If you have a catalogue with many products and the import is slowed down, you can still deactivate the real-time sync from the SYNC tab of the module and still use the cron solution.

In the second case Fastbay will NOT be activated during import. Therefore, an alignment to eBay “asynchronous” (not in real time) will be necessary at the end of the import, using the manual sync or the synchronization crons. For the setting of the latter, you can refer to the dedicated “cron” article in this documentation, Fastbay allows you an optimised update mode via cron to send only an incremental list of modified products to eBay, by checking the update date of the products themselves (“reviselastimport” cron).