UPDATE 2019-07-11

Here's an amended solution which works for scenarios when there are multiple values in the JSON: https://dbfiddle.uk/?rdbms=sqlserver_2017&fiddle=1fde45dfb604b2d5540c56f6c17a822d

update a
set details = JSON_MODIFY(details, '$[' + x.[key] + '].FieldValue', ename)
from dbo.tblUissAssignments a
CROSS APPLY OPENJSON (details, '$') x
CROSS APPLY OPENJSON (x.Value)
    WITH (CaptionName NVARCHAR(100),
        FieldName NVARCHAR(100),
        FieldType NVARCHAR(15),
        FieldValue NVARCHAR(50)) AS P
WHERE a.ssn = '000-00-0000'
and p.CaptionName = 'txtEname'    

This is similar to my original answer (see below). However:

• We now have 2 cross apply statements. The first is used to split the JSON array into elements, so we get a key (index) and value (JSON object as a string), as documented here: https://learn.microsoft.com/en-us/sql/t-sql/functions/openjson-transact-sql?view=sql-server-2017#path
• The second does what your original CROSS APPLY did, only acting on the single array element.
• We use the [key] returned by the first cross apply to target the item in the array that we wish to update in our JSON_MODIFY statement.

NB: If it's possible for your JSON array to contain multiple objects that need updating, the best solution I can think of is to put the above statement into a loop; since 1 update will only update 1 index on a given JSON. Here's an example: https://dbfiddle.uk/?rdbms=sqlserver_2017&fiddle=120d2ac7dd3a024e5e503a5f64b0089e

declare @doWhileTrueFlag bit = 1

while (@doWhileTrueFlag = 1)
begin

    update a
    set details = JSON_MODIFY(details, '$[' + x.[key] + '].FieldValue', ename)
    from dbo.tblUissAssignments a
    CROSS APPLY OPENJSON (details, '$') x
    CROSS APPLY OPENJSON (x.Value)
        WITH (CaptionName NVARCHAR(100),
            FieldName NVARCHAR(100),
            FieldType NVARCHAR(15),
            FieldValue NVARCHAR(50)) AS P
    WHERE a.ssn = '000-00-0000'
    and p.CaptionName = 'txtEname'    
    and p.FieldValue != ename --if it's already got the correct value, don't update it again

    set @doWhileTrueFlag = case when @@RowCount > 0 then 1 else 0 end

end

ORIGINAL ANSWER

Try this: https://dbfiddle.uk/?rdbms=sqlserver_2017&fiddle=b7b4d075cac6cd46239561ddb992ac90

update a
set details = JSON_MODIFY(details, '$[0].FieldValue', ename)
from dbo.tblUissAssignments a
cross apply
    OPENJSON (details)
        WITH (CaptionName NVARCHAR(100),
              FieldName NVARCHAR(100),
              FieldType NVARCHAR(15),
              FieldValue NVARCHAR(50)) AS P
where a.ssn = '000-00-0000'
and p.captionname = 'txtEname' 

More info on the JSON_MODIFY method here: https://learn.microsoft.com/en-us/sql/t-sql/functions/json-modify-transact-sql?view=sql-server-2017

The subtle bit is that you're updating a json array containing a json object; not a single object. For that you have to include the index on the root element. See this post for some useful info on JsonPath if you're unfamiliar: https://support.smartbear.com/alertsite/docs/monitors/api/endpoint/jsonpath.html

Regarding scenarios where there's multiple items in the array, ideally we'd use a filter expression, such as this:

update a
set details = JSON_MODIFY(details, '$[?(@.CaptionName == ''txtEname'')].FieldValue', ename)
from dbo.tblUissAssignments a
where a.ssn = '000-00-0000'

Sadly MS SQL doesn't yet support these (see this excellent post: https://modern-sql.com/blog/2017-06/whats-new-in-sql-2016)

As such, I think we need to apply a nasty hack. Two such approaches spring to mind:

• Implement a loop to iterate through all matches
• Convert from JSON to some other type, then convert back to JSON afterwards

I'll think on these / whether there's something cleaner, since neither sits comfortably at present...

If I understand your question, then one possible approach (if you use SQL Server 2017+) is to use OPENJSON() and string manipulations with STRING_AGG():

Table:

CREATE TABLE #Data (
   id int, 
   ssn varchar(12), 
   ename varchar(40),
   details nvarchar(max)
)
INSERT INTO #Data
   (id, ssn, ename, details)
VALUES
   (1, '000-00-0000', 'stackoverflow1', N'[{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":""}, {"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":""}]'),
   (2, '000-00-0000', 'stackoverflow2', N'[{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":""}, {"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":""}]')

Statement:

SELECT
   d.id, d.ssn, d.ename,
   CONCAT(N'[', STRING_AGG(JSON_MODIFY(j.[value], '$.FieldValue', ename), ','), N']') AS details
FROM #Data d
CROSS APPLY OPENJSON (d.details) j
WHERE JSON_VALUE(j.[value], '$.CaptionName') = N'txtEname' AND (d.ssn = '000-00-0000')   
GROUP BY d.id, d.ssn, d.ename

Output:

id  ssn         ename           details
1   000-00-0000 stackoverflow1  [{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":"stackoverflow1"},{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":"stackoverflow1"}]
2   000-00-0000 stackoverflow2  [{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":"stackoverflow2"},{"CaptionName":"txtEname","FieldName":null,"FieldType":null,"FieldValue":"stackoverflow2"}]

For SQL Server 2016 you may use FOR XML PATH for string aggregation:

SELECT
   d.id, d.ssn, d.ename,
   CONCAT(N'[', STUFF(s.details, 1, 1, N''), N']') AS details
FROM #Data d
CROSS APPLY (
   SELECT CONCAT(N',', JSON_MODIFY(j.[value], '$.FieldValue', ename))
   FROM #Data
   CROSS APPLY OPENJSON (details) j
   WHERE 
      (JSON_VALUE(j.[value], '$.CaptionName') = N'txtEname') AND
      (ssn = '000-00-0000') AND
      (id = d.id) AND (d.ssn = ssn) AND (d.ename = ename)
   FOR XML PATH('')
) s(details)