Unless things have changed recently you won't be able to online check in on a BA ticket on the Aer Lingus Codeshare flights from DUB-LHR - ba.com won't allow you to check in for Aer Lingus operated flights and Aerlingus.com will only allow check in on Aer Lingus issued tickets, not BA ones ... nor will you be able to use the Aer Lingus self service machines I don't think - so it will be old fashioned queueing up at the desk.
You might be able to check in on BA.com for your Heathrow - Sydney flight 24 hours in advance of it's departure so you can select decent seats .. not sure if that would work given that you won't have checked in for the DUB-LHR flight before it but it might be worth a shot