Is there a way to copy/initialize values in a set
Until we see the actual code, we can only speculate but for what is worth, I'll do a guess.

Assuming that your code involves only a table and its columns (no @variables involved) and is something like:

UPDATE
    tablename
SET
    column_a = column_b, 
    column_b = 0 
WHERE 
    (condition) ;


It is valid SQL and 100% correct. Correct, meaning that after the update column_a will have the values that column_b had before the update and column_b will have 0, for all rows that pass the (condition).

SQL-Server follows this behaviour strictly and there is no reason to assume that it will deviate from the standard in a future release.

SQL works in sets and the updates are done "simultaneously" on the whole row. (they may not be done simultaneously but the result is the same as if they were.) You could change the code to this and the result will be (guaranteed) the same:

...
SET
    column_b = 0,
    column_a = column_b
...


In fact, the following is valid and equivalent SQL as well (although not yet implemented in SQL-Server);

UPDATE
    tablename
SET
    (column_a, column_b) = (column_b, 0) 
WHERE 
    (condition) ;


I haven't tested Azure but if it deviated from this basic SQL behaviour, it would better be documented somewhere.