What is the furthest distance in any direction from the centre of Australia to the coast?

Mainland speaking, approximately 2000 km.

The mainland, from it's most northern point to its most southern point, is approximately 3700 km long. The mainland, from its most eastern point to its most western point, is approximately 4000 km wide. Thus, 4000 km is the longest distance (coast to coast) on the mainland...and, from the center of that span to the coast would be approximately 2000 km.

This would be the best possible answer, as Australia has no official center.

http://www.ga.gov.au/scientific-topics/national-location-information/dimensions/continental-extremities
http://www.ga.gov.au/scientific-topics/national-location-information/dimensions/centre-of-australia-states-territories#fivecentres

Eromanga is the town furthest from the ocean in all of Australia.

http://www.studytoursqueensland.com.au/destinations/outback-&-western-downs.cfm