As long as you have suitable indexes in place this should work alright:
UPDATE table_a
SET
column_a_1 = (SELECT table_b.column_b_1
FROM table_b
WHERE table_b.user_name = table_a.user_name )
, column_a_2 = (SELECT table_b.column_b_2
FROM table_b
WHERE table_b.user_name = table_a.user_name )
WHERE
EXISTS (
SELECT *
FROM table_b
WHERE table_b.user_name = table_a.user_name
)
UPDATE in sqlite3 did not support a FROM clause for a long time, which made this a little more work than in other RDBMS. UPDATE FROM was implemented in SQLite 3.33 however (2020-08-14) as mentioned at: https://stackoverflow.com/a/63079219/895245
If performance is not satisfactory, another option might be to build up new rows for table_a using a select and join with table_a into a temporary table. Then delete the data from table_a and repopulate from the temporary.