No, this will not work. I believe this POE standard uses the non-data pins in a CAT5 or CAT5e cable such as suggested here and here.
Your other comment about it starting as an 8 pin cable but that they split later on explains the situation a lot better. The installer put in CAT5 cable but because cable is ‘expensive’ (the installer is a cheapskate and cheater) they split the cable into 2x100Mbit. This works, but is definitely not up to code or supported in any way.
My personal opinion, rip it out and replace it, do it correct. You don’t need to pull a single cable to every apartment/room. You can replace 1 cable, put a small (5 port for instance) switch in a central location and then branch from there. That should save in replacement cost (long cables are more expensive then a little 5p Gigabit switch) and upgrade everything to gigabit while you’re at it! This could also partially give you POE on the cables if you replace the ‘in-the-middle’ switch you are adding with a 5p Gigabit POE model for instance. Or you can passively inject from there.
But again, what you are trying to build right now, please don’t. This is in no way compliant to anything and will cause lots of problems and headaches over time, costing more then the materials and replacing them ever would have. Also, the mess you are now in will be even worse for the next guy again…