Most efficient way to update a column where Id exists in another table
I believe by doing an EXISTS on a subquery you may be using a sub-optimal way to accomplish this. Why not emulate an INNER JOIN directly like so:
update fiscal_year y
set y.process_date = now()
from owners o
where y.owner_id = o.owner_id
and y.process_date is null;

Having an index on (owner_id, process_date) for the fiscal_year table should help ensure the most optimal operations occur in this case too.
Your tables aren't big by any means, this process should be able to execute in minutes if not seconds, as long as everything is architected properly. I'd be curious what the above yields you (with the recommended index).