I would have used WEEKNUM in combination with a VLOOKUP.  The VLOOKUP should
be used with a hard-coded table where you fill in the weeknumber in column 1
and the corresponding month in column 2.

Typical formula can be as follows:

=VLOOKUP(A1, A2:B13, 2, True)

where your weeknumber is in cell A1, and range A2 through B13 contain the 12
weeks that define the start of a month.  This should do the trick.  Of
course, since the 1st of every month does not fall on the start of a week,
you will always have a gray area at the end of each month and start of each
month.  But that is the degree of precision that you have requested in your
question, thus cannot do any better.