This was what I did last season.

Take the South Line train (CityRail)
Dep: 8:58am Airport International Station Platform 1
Arr: 9:11am Central Station Platform 21

Take the Western Line train (CityRail)
Dep: 9:21am Central Station Platform 18
Arr: 9:47am Clyde Station Platform 5

Take the Carlingford Line train (CityRail)
Dep: 9:52am Clyde Station Platform 1
Arr: 9:55am Rosehill Station Platform 2

Walk to Rosehill Racecourse, about a furlong.