Other than being sent as convicts, what pull factors drew many Irish migrants to Australia?
First of all, please note that being sent somewhere as a convict is not a pull factor. That is, instead, a push factor. It is a reason to leave one's home country as opposed to a reason to come to some particular new place.
So when you look for pull factors about Australia, you have to ask "why would Irish migrants want to go to Australia" rather than asking "why would they leave Ireland." The main pull factor that brought people to Australia was the prospect of land. There was a whole huge continent that had no Europeans to own the land. This would have offerred the Irish good economic prospects. Other than that, the only real pull factor would have been the fact that Australia's culture would have been somewhat familiar to the Irish. They would be going to a place that was being colonized by the English as opposed to some other country where they would not know the language and customs.