Hi Kate
Trying to find out when an ancestor arrived can be extremely difficult. I have one or two who either swam here or were dropped from a space ship!
I don't know whether this will be of any help and you may already have it - but just in case -
A young woman named Jane O'Brien aged 22 arrived Melbourne via the ship "Constance" in December 1852. (From Victorian Public Records Office on line passenger lists).
Do you have a copy of Thomas Fanning's death certificate. If so it may state how long he lived in the colony which will provide the approximate of year of his arrival.That can narrow down the search considerably. Old newspapers of the time also often recorded the names of passengers arriving and the ship they came on. Good luck - Trish