That cable you purchased should be fine Bash. And yes, you
must have an active adapter for Eyefinity. Passive adapters will work for 3 separate monitors that are not creating a single desktop, but for one large Eyefinity resolution they must be active, either single or dual link. Single link for 1920 x 1200 or less per monitor.
I forgot to mention the 24 inch that i will be purchasing has a displayport and my video card is ati 5770 with displayport.
I don't have a concrete answer for you but, I have wondered where the bottom end is for 1080p Eyefinity and Aces High specifically/only in regards to graphic cards. My gut feeling after lots of research is ... that single 5770 won't cut it. Yes, it is Eyefinity capable, but for most other games it doesn't have the power to drive 3 monitors at playable Eyefinity resolutions. Maybe, for Aces High and it would be good to know for sure. Crossfired would likely be the best bet.
It would be good to get a guinea pig to say for sure and it looks like you're it.