You could use an updatable table expression making use of a window aggregate:

update t 
  set Keys = case when k is null then 'None' end
from (
  select *, Max(keys) over(partition by Building_Name) k
  from t
)t
where Janitor = 'No_Name';

I dont know the exact DBMS you're using, but the idea is generally the same. You can use aggregation function like MAX() on Keys to generate list of buildings to update to None.

select "Building name"
from No_Name
group by "Building name"
having max(keys) <> NULL

this query will give you all building that have all NULL values in keys, then you can incorporate it in Update statement.

You can try going with an EXISTS expression.

UPDATE tab
SET Keys = 'None'
WHERE Janitor = 'No_Name'
  AND NOT EXISTS(SELECT 1 
                 FROM tab t2 
                 WHERE tab.BuildingName = t2.BuildingName AND t2.Keys IS NOT NULL);

Output:

Check the demo here .