How is ATC handed for private rural airstrips?
I can’t speak for Australia, but in the United States there are thousands of uncontrolled airports throughout the country, both privately and publicly operated.  Larger, better equipped public uncontrolled airports have good facilities such as paved runways and taxiways, instrument approaches, automated weather reporting stations, pilot controlled airport lighting, FBOs, fuel and maintenance depots, etc. but many uncontrolled airports are simply grass, gravel or dirt strips with little or no facilities, particularly in back country or mountainous areas.

Typically the decision to go into these kinds of airports is at the pilot’s discretion and the pilot alone must make the final call as to whether they can safely take off and land at these locations.  Bush pilots departing from said locations will typically carry small handheld weather stations used by mountaineers for determining local atmospheric conditions, winds, etc.  Remote airports will use a standard left hand traffic pattern flown at 1000’ AGL, but many do not due to nearby terrain and are one way in/out.  If a CTAF is not published for the airport, pilots will use the MULTICOM frequency of 122.9 MHz to announce their position and intentions.

Condition of the airport and runway are often not monitored and it is up to the pilot to determine if they can be used.  Arriving traffic will often make a low, slow pass over the airfield to determine runway condition and look for obstacles which could impede landing.

All types of aircraft can and do use such airports but most often the aircraft operating from them are specially build bush aircraft with good STOL flight characteristics plus beefy landing gear and tires for operating from unpaved strips or off field operations.