I have a CAT5e cable with only 3 pairs (6 wires working) and need to do POE over it

A starting note: If one pair of your cable is damaged, and this happened through some kind of physical abuse to the cable (or if you don't know how it happened), there is a very good chance that other pairs are also damaged, and you just don't have the equipment to detect it. If so, anything you try to with this cable is likely to work poorly, if at all.

Regarding data: in normal use, the orange pair would be carrying data, so you will need a nonstandard pinout if you want to use this cable at all. The standard pinouts are called T-568A and T-568B:

Note that the pins used for data are 1-2 and 3-6, and each of those pairs MUST be kept together (that is, 1-2 must be the same color pair, and 3-6 must be the same color pair.) For a normal "straight-through" cable, you would use either A or B (usually B, in my experience) on both ends.

One possible pinout that you could use with your cable: WGreen-Green-WBlue-Orange-WOrange-Blue-WBrown-Brown. From T-568A, this swaps orange with blue, keeps 1-2 and 3-6 paired as is required, and moves the orange pair into the unused center position (where it will remain harmlessly unused even if you wire both ends backwards, which is very easy to do by mistake.)

In order to use PoE with this setup, it appears that you will need what's called a "Mode A" PoE injector ("power sourcing equipment"). Quoting from https://en.wikipedia.org/wiki/Power_over_Ethernet -- "In mode A, pins 1 and 2 ... form one side of the 48 V DC, and pins 3 and 6 ... form the other side. These are the same two pairs used for data transmission in 10BASE-T and 100BASE-TX, allowing the provision of both power and data over only two pairs in such networks." This will be necessary in your situation, since all other PoE modes require all four pairs to be working.

Unfortunately, I can't find any documentation on whether your switch uses Mode A or Mode B for PoE. (Luckily, the PoE standard apparently requires the AP to accept either.) Wikipedia and some other sources suggest that Mode B is typical for injectors, and Mode A is typical for devices like switches, but this is far from guaranteed. If you already have the equipment, your best bet may be to try it and see; however it's not impossible that shorts in the cable could cause damage to your devices, so try at your own risk!