postgresql trigger for updating multiple columns
You need to change your query to become like this:
UPDATE res_partner
SET x_street1=res_partner_address.street
FROM res_partner_address
WHERE res_partner.id = NEW.partner_id;
You should make use of the NEW row. This refers to the inserted row in case of INSERT operation. It also refers to the updated row in case of UPDATE operation.
By the way, you don't need to create the trigger for DELETE operation.