So, while I still think you're setting yourself up if you're not incredibly careful, here's one of the most cautious ways to approach this to make sure the data can't be double-touched between the select and the updates.

Note, I switched the sample data over to a temp table so you could actually do an update.

IF OBJECT_ID ('tempdb..#SampleData') IS NOT NULL

DROP TABLE #SampleDAta

CREATE TABLE #SampleData (PERSON INT ,dt DATETIME,[PAYTYPE] VARCHAR(20) ,[ID] INT ,[DOW] VARCHAR(20))

INSERT INTO #SampleData

SELECT 12913,'07/21/2014','Regular',101,'Monday'

UNION ALL SELECT 12913,'07/22/2014','Regular',102,'Tuesday'

UNION ALL SELECT 12913,'07/23/2014','Regular',103,'Wednesday'

UNION ALL SELECT 12913,'07/24/2014','Regular',104,'Thursday'

UNION ALL SELECT 12913,'07/25/2014','Regular',105,'Friday'

UNION ALL SELECT 12913,'07/26/2014','Regular',104,'Saturday'

UNION ALL SELECT 12913,'07/27/2014','Regular',105,'Sunday'

UNION ALL SELECT 12913,'07/28/2014','Regular',101,'Monday'

UNION ALL SELECT 12913,'07/29/2014','Regular',102,'Tuesday'

UNION ALL SELECT 12913,'07/30/2014','Regular',103,'Wednesday'

UNION ALL SELECT 12913,'07/31/2014','Regular',104,'Thursday'

UNION ALL SELECT 12913,'08/01/2014','Regular',105,'Friday'

DECLARE @RunDate DATETIME

SET @RunDate = '20140730'

DECLARE @Result TABLE

(PERSON INT ,dt DATETIME,[PAYTYPE] VARCHAR(20) ,[ID] INT )

SELECT

dt,

DATEPART( week, dt - 1) -- -1 because standard weeks start on sunday)

FROM

#SampleData

WHERE

DATEPART( week, dt - 1) = DATEPART( week, @RunDate - 1)

AND dt <= @RunDate

UPDATE #SampleData

SETPayType = 'Regular2'

OUTPUT

inserted.Person,

Inserted.dt,

Inserted.PayType,

inserted.id

INTO

@Result

WHERE

PAyType = 'Regular'

AND DATEPART( week, dt - 1) = DATEPART( week, @RunDate - 1) -- -1 because standard weeks start on sunday)

AND dt <= @RunDate

SELECT * FROM @Result

Let me know if there's any confusion or concerns. Be aware that the DATEPART component is going to die a horrid death because of no indexing. I'd recommend you turn that into a computed persisted column on the table and index it.