How to rank in postgres query

By placing the rank() function in the subselect and not specifying a PARTITION BY in the over clause or any predicate in that subselect, your query is asking to produce a rank over the entire url_info table ordered by pub_date. This is likely why it ran so long as to rank over all of url_info, Pg must sort the entire table by pub_date, which will take a while if the table is very large.

It appears you want to generate a rank for just the set of records selected by the where clause, in which case, all you need do is eliminate the subselect and the rank function is implicitly over the set of records matching that predicate.

select 
  cluster_id
 ,feed_id
 ,pub_date
 ,rank() over (order by pub_date asc) as rank
from url_info
where cluster_id = 9876 and feed_id = 1234;

If what you really wanted was the rank within the cluster, regardless of the feed_id, you can rank in a subselect which filters to that cluster:

select ranked.*
from (
  select 
    cluster_id
   ,feed_id
   ,pub_date
   ,rank() over (order by pub_date asc) as rank
  from url_info
  where cluster_id = 9876
) as ranked
where feed_id = 1234;