UPDATE

    Table_A

SET

    Table_A.col1 = Table_B.col1,

    Table_A.col2 = Table_B.col2

FROM

    Some_Table AS Table_A

    INNER JOIN Other_Table AS Table_B

        ON Table_A.id = Table_B.id

WHERE

    Table_A.col3 = 'cool'

Your query does not work because you have no FROM clause that specifies the tables you are aliasing via A/B.

Please try using the following:

UPDATE A

    SET A.NAME = B.NAME

FROM TableNameA A, TableNameB B

WHERE A.ID = B.ID

Personally I prefer to use more explicit join syntax for clarity i.e.

UPDATE A

    SET A.NAME = B.NAME

FROM TableNameA A

    INNER JOIN TableName B ON

        A.ID = B.ID

  UPDATE a

        SET    a.CustomTableItemID = b.CustomTableItemID

        FROM   @TempTable AS a

               INNER JOIN dbo.tbm_cti_CustomTableItem AS b ON a.CustomTableItemGUID = b.CustomTableItemGUID;