SQL Server ignores trailing whitespace when comparing strings, so ' ' = ''. Just use the following query for your update
UPDATE table
SET col1 = NULL
WHERE col1 = ''
NULL values in your table will stay NULL, and col1s with any number on space only characters will be changed to NULL.
If you want to do it during your copy from one table to another, use this:
INSERT INTO newtable ( col1, othercolumn )
SELECT
NULLIF(col1, ''),
othercolumn
FROM table