Since you can t update subqueries with row_number, you ll have to calculate the row number in the set part of the update. At first I tried this:

update voyage_port a
set a.port_seq = (
  select 
    row_number() over (partition by voyage_id order by arrival_date)
  from voyage_port b
  where b.voyage_port_id = a.voyage_port_id
)

But that doesn t work, because the subquery only selects one row, and then the row_number() is always 1. Using another subquery allows a meaningful result:

update voyage_port a
set a.port_seq = (
  select c.rn
  from (
      select 
        voyage_port_id
      , row_number() over (partition by voyage_id 
            order by arrival_date) as rn
      from voyage_port b
   ) c
  where c.voyage_port_id = a.voyage_port_id
)

It works, but more complex than I d expect for this task.

You can update some views, but there are restrictions and one is that the view must not contain analytic functions. See SQL Language Reference on UPDATE and search for first occurence of "analytic".

This will work, provided no voyage visits more than one port on the same day (or the dates include a time component that makes them unique):

update voyage_port vp
set vp.port_seq =
( select count(*)
  from voyage_port vp2
  where vp2.voyage_id = vp.voyage_id
  and vp2.arrival_date <= vp.arrival_date
)

I think this handles the case where a voyage visits more than 1 port per day and there is no time component (though the sequence of ports visited on the same day is then arbitrary):

update voyage_port vp
set vp.port_seq =
( select count(*)
  from voyage_port vp2
  where vp2.voyage_id = vp.voyage_id
  and (vp2.arrival_date <= vp.arrival_date)
  or (   vp2.arrival_date = vp.arrival_date 
     and vp2.voyage_port_id <= vp.voyage_port_id
     )
)

Don t think you can update a derived table, I d rewrite as:

update voyage_port
set port_seq = t.new_seq
from
voyage_port p
inner join
  (select voyage_port_id, voyage_id, arrival_date, port_seq,
   row_number() over (partition by voyage_id order by arrival_date) as new_seq
   from voyage_port) t
on p.voyage_port_id = t.voyage_port_id

The first token after the UPDATE should be the name of the table to update, then your columns-to-update. I m not sure what you are trying to achieve with the select statement where it is, but you can update the result set from the select legally.
A version of the sql, guessing what you have in mind, might look like...

update voyage_port t
set t.port_seq = (<select statement that generates new value of port_seq>)

NOTE: to use a select statement to set a value like this you must make sure only 1 row will be returned from the select !

EDIT : modified statement above to reflect what I was trying to explain. The question has been answered very nicely by Andomar above