So recently I've been getting into Van Halen, and naturally I want to use my tremolo (my guitar has its trem blocked because it does not stay in tune at all). Now on my guitar: http://www.musiciansuniverse.com.au/Uploads/Images/150(1).jpg
the tremolo is an Ibanez FAT-10: http://lh5.ggpht.com/_MlLIWOFjyc4/S6Br5MyQ1MI/AAAAAAAACTQ/_g4O8b46br0/s400/P1080512.JPG
which floats above the body with two posts (as you can see). It's almost the same as a Floyd Rose but without locking the strings. I know (from asking on this very forum) that to put in a Floyd Rose would require a lot of routing, which I'm not willing to do. However, would an original floyd without the fine tuners need the routing? I would check myself whether it would fit with the two posts but I couldn't find the dimensions anywhere. Also would I need a locking nut, because if so then this whole thing is screwed as I don't have room for one on my guitars neck and I'm not getting a new one. I don't want to go crazy on it, just a dive bomb every now and then and a bit of vibrato.

The first thing you can do is get some locking tuners and a nice nut that is cut properly. Those 2 things should go a loooooong way to keeping the guitar in tune. I would do that before you replace the trem you will probably be surprised. Even if you swapped to a different trem you would want to do those mods if you can't get a locking nut.
Your trem is more like a vintage style trem, it would need routing to make it fully floating.