Hey,
It is possible to process AAS via web activity through REST API.
Git hub repo for code:
https://github.com/NandanHegde15/Azure-DataFactory-Generic-Pipelines/tree/main/Azure%20Analysis%20Service/ProcessAzureAS
W.r.t the other question: it basically depends on your data ingestion process.
Ideally once the data ingestion is successful,then you process the cube.
So in case if you use an ADF to ingest the data
It would be better to leverage that for cube processing.