ExPat, Sydney diving is great but to experience it at its best you really need either a car or a local to drive you to the various sites.
Sounds like you are keen to see Weedy's? When I think Weedy I think Kurnell or Jervis Bay ... if you had 5 full days and were particularly keen to dive, you might like to go to JB which is a few hours south of Sydney. Take a look at the live aboard Oceantrek http://www.oceantrek....au/frames.html for a 2 or 3 day trip? It is an older boat, not flash (and an old style website) but a good stable platform with a very friendly and knowledgeable crew.
Re Weedy's in Sydney, sightings for the past few years have been down (Sydney is at the top of their range) but you can on occasion see them at Shelly (Manly). I've never known one to make it to Ship Rock but I guess it may have occurred .. the Sydney site with best Weedy prospects would be the Kurnell dive sites (The Leap, The Steps, The Monuments, Inscription Point....)
Michael McFadyen's web site, www.michaelmcfadyenscuba.info is a great reference site for Sydney diving.
If you are on facebook you can also view my dive clubs page (need to be a club member to join the group) https://www.facebook...08786489161552/ where amongst the posts you will find fresh dive reports that will give you a feel for conditions closer to the date (although not many reports at the moment as it has been raining all week with associated wash out).