creating a trigger after update exclude a column?
I believe your code is missing or is typo'ed

CREATE TRIGGER [dbo].[product_trig]
  ON dbo.product
  AFTER UPDATE
AS -- you miss this 'as'
BEGIN -- you miss this 'begin'
    SET NOCOUNT ON;
    IF NOT UPDATE (to_sync) 
      BEGIN
          UPDATE dbo.product 
              SET  to_sync  = 'TRUE'
          FROM dbo.product INNER JOIN INSERTED i ON dbo.product.NAME = i.NAME
      END
END