You have three options what to do after Sitecore Publishing Service is done:
You can hook on publishEndResultBatch pipeline. You can then work with these results and update any third-party systems or features that may need to know about the changes to items.
With the Sitecore Publishing Service you have new event publishingservice:publishend which you can use. There are no remote events anymore for publishing service as service is already done remotely. See chapter 4.7 from installation manual referenced below.
When you
Publish all Items
, you can check Clear all data caches checkbox. This will clear the data level caches that contain a reference to the items that are published. See chapter 7.4 from installation manual referenced below. - For your scenario not goodthis won't work as expected as you have implemented custom cache.
For your scenario, definitely the 1. option is best. Hook on publishEndResultBatch pipeline and then work with these results. Remove items from your cache by comparing ids with ids that come as a result of this pipeline.
More information can be found in documentation: