Insert Array of JSON into Postgres Table
PART1 - INSERTING

You don't have to use unnest() but jsonb_array_elements() and add square brackets to the JSON data structure. I sugget you using a JSON validator website like JSONlint to test the correctness of your JSON data.

This codes inserts 3 new records in inputtable:

WITH json_array AS (
    SELECT 1 AS user_id, 
           2 AS chart_id,
           jsonb_array_elements('
               [
                 {
                    "col1": "a",
                    "col2": 1,
                    "col3": 1,
                    "col4": "one"
                 }, {
                    "col1": "b",
                    "col2": 2,
                    "col3": 2,
                    "col4": "two"
                 }, {
                    "col1": "c",
                    "col2": 3,
                    "col3": 3,
                    "col4": "three"
                }
               ]'::jsonb) AS properties
)
INSERT INTO inputtable (user_id, chart_id, properties) 
SELECT * FROM json_array


PART2 - UPDATING

In order to update you must specify data_point_id values, so you have to know them a priori.

This works perfectly, but probably there are other naive solutions:

WITH update_table AS(
   SELECT unnest(ARRAY[1, 2, 3]) AS data_point_id,
          jsonb_array_elements('
            [
              {
                "col1": "a",
                "col2": 6,
                "col3": 7,
                "col4": "eight"
              }, {
                "col1": "b",
                "col2": 10,
                "col3": 11,
                "col4": "twelve"
              }, {
                "col1": "c",
                "col2": 3,
                "col3": 3,
                "col4": "new"
              }
           ]'::jsonb) AS properties 
    FROM inputtable
)
UPDATE inputtable 
SET properties = update_table.properties
FROM update_table
WHERE inputtable.data_point_id = update_table.data_point_id