Get the Archival Process report

short: no. 
"Data for processes that have completed (which are no longer needed for auditing or reporting purposes) must be archived regularly. Archiving a process removes it from being used by the process analytics engine."
see here.
https://docs.appian.com/suite/help/22.3/Managing_Process_Archives.html

the sense of a archiving a process is a kind of "clean up". an admin can reopen the archive and so on, but not like you would need it as a report.

is there any way that we can capture the process start date and end date of archival processes?

is there any way that we can capture the process start date and end date of archival processes?

again, not like you would need it. Appian states itself:

Unknown said:

(which are no longer needed for auditing or reporting purposes)

and you try to do a reporting on archived processes, which is not matching with the purpose of "archiving a process".