I never had troubles with the solo duties (except for the lvl 15 fight in the beta, because that one was just ridiculous..), and I did them on pretty much any role so far.
Pay attention to what NPCs say, in the middle of the screen AND in your chat window, they usually tell you what to do (which might be different from what is intuitive at the moment).

For this particular fight, I did it on my BLM. I just killed them in the order of: WHM -> BRD -> PLD/WAR (whichever kept interrupting me at the time). By then I could give Alisiae aether and/or they regenerated and I started with the WHM again.
Alisiae never even dropped below 90% HP for me. There are a lot of AoEs that get targeted at you / your immidiate sourroundings, so make sure to stay away from her except for the short moment at which you need to run over and interact with her.