Oracle standby databases monitoring
There's a few ways to determine if MRP is running.  One is to query open_mode from v$database.  If MRP is running, you will get:

READ ONLY WITH APPLY


If not, you will get:

READ ONLY


You can run the following query which will show what MRP processes are running:

select process,status,sequence# 
from v$managed_standby;


Look for the process MRPn

You could look in V$DATAGUARD_STATUS.MESSAGE for various things, including:

MRP0: Background Managed Standby Recovery process started
Managed Standby Recovery Canceled


Another way is on the primary, to check

select recovery_mode from v$archive_dest_status where dest_id = n;


If it's IDLE, it's not doing anything.  With real-time apply enabled, it returns MANAGED REAL TIME APPLY.