Yes, Intercity busses go everywhere you'd ever want to go, pretty much daily (several times a day on the popular routes). There's also a number of often-cheaper more regional alternatives. Visitor centers in any town should be able to tell you about the options and sell tickets. (Although since you'll be in the country for a number of months, it might be worth getting a FlexiPass for cheaper bus travel with Intercity instead.)
I spent extensive time in both countries and loved it, but with your given timeframe I'd also go for working 4 months in New Zealand and travelling a month in Australia, especially if you love beautiful landscapes and like less tourists.
Queenstown is definitely a good place to work during winter (I've done it myself a couple of years ago), but keep in mind that if you arrive in July the season is almost over. If you want to work for the whole 4 months, try and get there at the beginning/middle of May.
Besides that, public transport is good in N.Z. as Sander mentioned, but in Queentown you will also sure meet other backpackers with cars/vans who do trips on their days off and offer you to come along. Quite a friendly and fun community down there during winter season !