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I just had a look with Yellow Pages on the iPad and it seems that there is no real close bottle shops.
The closest that it gave was on the corner of Hunter & Elizabeth streets which I think would take between 15 - 20 minutes slow walk, maybe less.
There are a few close pubs that may have a bottle shop that might be able to sell you some.
Voyager of the Seas - Fremantle - Singapore, March 2014
Rhapsody of the Seas - South Pacific, November 2013
Rhapsody of the Seas - South Pacific, April 2013
Rhapsody of the Seas - South Pacific, February 2013
Sun Princess - Fiji, March 2012
Princess Elite, 2 Star Mariner (HAL) Way back, Fiji (Blue Lagoon), Queensland coast (NCL), (Fairstar), Islands, NZ, Tonga, Solomons.
2004 Pacific Princess (Fiji, Vanuatu,New Caledonia)
2006 Star Princess (Darwin to Bangkok)
2008 Diamond Princess (Sydney to Auckland)
2009 Dawn Princess(Cherry Blossom, Asia, Japan, PNG,)
2009 Diamond Princess Bangkok to Sydney
2010 Diamond Princess Singapore to Sydney
2011 Volendam Sth Pacific
2011 Radiance of the Seas Long Dinner Party
2012 Dawn Princess Mel/Syd
2012 Dawn Princess Syd/Brisbane
2012 Aegean Odyysey Instanbul to Venice
2012 Radiance of the Seas
2013 Ocean Princess Honolulu/Tahiti/Tonga
2013 Celebrity Solstice New Zealand
2014 Voyager of the Seas
2014 Azamara Journey India to Athens via Suez Canal
Careful with RCL though. When I boarded Voyager 5th Jan they had a massive sign just before you got to security stating "No drinks (including alcohol) could be brought onboard." There was a whole lot of verbiage describing those rules.
I know they only updated their policy a couple of months back, but you may not have an easy go of it getting any onboard.