It's probably a reference to 'Sky' Masterson's song (in Guys and Dolls: 'Luck be a lady' (in which the first words are 'They call you Lady Luck' - the goddess of good fortune - the Latin goddess was just 'Fortuna', but English translators of songs and prayers addressed to her had to supply the 'Lady'). After calling her Lady Luck the song then uses the brilliant (I think) pun 'you have an un-lady-like way of running out'.
I don't know about the 'Old'. The Bank of England (in Threadneedle Street, London) is known as 'the Old Lady of Threadneedle Street'. 'The old lady' is also an informal (and rather dated) way of saying 'my wife'. But I very much doubt if either of these explains the SATC title'.