SQL set values of one column equal to values of another column in the same table

Sounds like you're working in just one table so something like this:

update your_table
set B = A
where B is null

I don't think that other example is what you're looking for. If you're just updating one column from another column in the same table you should be able to use something like this.

update some_table set null_column = not_null_column where null_column is null

I would do it this way:

UPDATE YourTable SET B = COALESCE(B, A);

COALESCE is a function that returns its first non-null argument.

In this example, if B on a given row is not null, the update is a no-op.

If B is null, the COALESCE skips it and uses A instead.

UPDATE YourTable
SET ColumnB=ColumnA
WHERE
ColumnB IS NULL 
AND ColumnA IS NOT NULL