Generally speaking, no, you will not cycle with old water. There can be other benefits though, in your case.
Nitrifying bacteria live on surfaces covered by water. They adhere to what is termed a biofilm, which is a sticky substance that holds bacteria, microscopic creatures, algae, etc. This is why transferring filter media or the entire filter is useful. With the water flow, there will usually (but not always) be more bacteria on filter media. If you could run the sponge filter in the main tank for a few days, it would likely be partially colonized with bacteria.
As for the other benefits. If you are moving eggs or fry, or fish in general as to a hospital tank, using some or all of the "old" tank's water can lessen the shock caused by differing parameters and biological chemistry. When dealing with healthy and mature fish, given all the "bad" stuff in the water it can be more detrimental than beneficial.
So in your case, as you are intending to transfer fry, I would use all water from the old tank and after the fry are transferred, do partial water changes but maintain close parameters (GH, pH, temp).
Byron.