Most participants will fly into Dunedin "International" Airport. The airport does indeed operate a small number of flights to Australia. However it is likely that you will enter the country at either Auckland or Christchurch and get a domestic flight to Dunedin. The airport is quite distant from the city (about 25km) and a taxi fare is quite expensive, around $60.
A cheaper alternative is to use an airport shuttle to St. Margaret's College. You can catch a regular shuttle bus or a supershuttle. The price of the regular shuttle is between $15 and $60 depending on how full it is. The supershuttle costs $27 and you have to book it in advance at www.supershuttle.co.nz.