An option is to patch the secondary node of AOAG clusters up to the latest patch, then the next month the business agree to schedule a fail over out of hours.. I can then patch the new secondary (old primary).
When you plan for patch update you must plan it for all replicas in AG to keep solution really Always On, it's one of core advantage of Availability Groups that server maintenance can be done without downtime.
In your approach your breaking the term of Always-on business continuity, for example, when you update secondary replica, leave primary replica without updating. For some-reason you wanted to fail-over to the secondary, that moment on-wards it may not fail-back over to the original primary again - one of the reason that the databases always upgraded to newer version but never be downgraded, there must be some workaround (scripting out) to downgrade the DB version, in this case when you cannot fail-back over to primary replica it's not Always On, thus, it's recommended to schedule patch update for all replicas together with recommended order..
Although, there is no down-time, the better time to perform patch updated is during less overload on the servers. Before proceeding, you may want to consider (if not configured already) Node and fail-share majority quorum configuration at WSFC as it's recommended when there are even number of nodes in WSFC, so the file-share witness maintain a vote to keep healthy quorum and cluster resources healthy (listeners) especially when secondary node is offline which normal during patch update.
Following query would be helpful to see the synchronization health (it's essential before and after doing patch update), some info cannot be available in the availability group dashboard but you can get it though DMVs (as follows):
ag.name as GroupName,
(redo_queue_size / 1024.0) as redo_queue_size_MB,
from sys.databases as db
left outer join sys.availability_databases_cluster as AGDB on db.group_database_id = AGDB.group_database_id
left outer join sys.dm_hadr_database_replica_states as DBRepStats on db.group_database_id = DBRepStats.group_database_id
left outer join sys.availability_replicas as Rep on DBRepStats.group_id = Rep.group_id and DBRepStats.replica_id = Rep.replica_id
left outer join sys.availability_groups as AG on DBRepStats.group_id = AG.group_id
where db.database_id > 4