UPDATE with join condition on matching words in columns of another table

The key here is that Column_1, represents three possible values for the JOIN. So what you want to use is string_to_array() (so long as those values are comma-separated and can not themselves include a comma).

Run this query,

SELECT id_number, string_to_array(column_1, ',') AS column_1
FROM table_a;
 id_number |       column_1        
-----------+-----------------------
       123 | {foo,bar,baz}
       456 | {qux,quux,quuz}
       789 | {corge,grault,garply}
       101 | {qux,bar,grault}

Now, we can run our UPDATE using = ANY(),

UPDATE table_b
SET id_number = A.id_number
FROM (
  SELECT id_number, string_to_array(column_1, ',') AS column_1
  FROM table_a
) AS A
WHERE table_b.column_1 = ANY(A.column_1)
  AND table_b.column_2 = ANY(A.column_1);

You can alternatively use <@

WHERE ARRAY[table_b.column_1, table_b.column_2] <@ A.column_1;

That even makes it a bit more compact..

UPDATE table_b
  SET id_number = A.id_number
FROM table_a AS A                              
  WHERE ARRAY[table_b.column_1, table_b.column_2] <@ string_to_array(A.column_1, ',');

Try this:

update table_b
set id_number = (select id_number
                 from table_a
                 where table_a.Column_1 like '%' || table_b.Column_1 || '%'
                       OR table_a.Column_1 like '%' || table_b.Column_2 || '%'
                 limit 1)
;

It can be another solutions by converting Column_1 into an array, but this is so clear.

Notice I'm limiting the search to 1 record, just in case the text appears in more than one column_1 of Table_A.

As Evan Carroll has pointed out in the comments sections, I'd remark that this code updates the whole table.

Check it here: http://rextester.com/MUL4593

Assuming Postgres 9.6, performance is relevant, big tables, "words" composed of characters, no whitespace or punctuation, no stemming or stop words, no phrases, all columns NOT NULL.

Full Text search backed by an index should be among the fastest solutions:

UPDATE table_b b
SET    id_number = a.id_number 
FROM   table_a a
WHERE  to_tsvector('simple', a.column_1)
    @@ plainto_tsquery('simple', concat_ws(' ', b.column_1, b.column_2))
AND    b.id_number = a.id_number;  -- prevent empty UPDATEs

With a matching expression index on a.column_1:

CREATE INDEX table_a_column_1_idx ON table_a USING GIN (to_tsvector('simple', column_1));