You can do that using an UPDATE statement after you addedTo add the new column use:
ALTER TABLE the_table ADD COLUMN id_int integer;
To populate the new column you need an UPDATE statement.
I am assuming you have a primary key column named pk_column
in your table. Obviously you need to replace that with the actual primary key column in your table.
update the_table
set id_int = t.rn
from (
select pk_column,
dense_rank() over (order by id_string) as rn
from the_table
) t
where the_table.pk_column = t.pk_column;
If you really have a table without a primary key (why?), you can use the built-in ctid
instead:
update the_table
set id_int = t.rn
from (
select ctid as id_,
dense_rank() over (order by id_string) as rn
from the_table
) t
where the_table.ctid = t.id_;