The earliest-recorded use of the phrase was in 1832 in a book by Captain Marryat - who wrote 'Midshipman Easy' etc. Whether that was where the phrase actually originated is uncertain.
Incitatus, I have always understood this phrase to be connected with Sundays being a day off work (we didn't used to get Saterdays off). Hence a month of Sundays would be 28-31 Sundays running consecutively. The significance being that on this day you wouldn't have your regular job to do. It might be translated as; If I had nothing at all to do for a whole month I still wouldn't get it finished.
Ta, Hamish