So I called Apple and opened a case with them. They told me that the AirPort card is part of the motherboard and it's not a user-installable part, which is why there's no existing part number for it. So I hate to be the bearer of bad news, but it appears that this guide is either wrong, or is only for a Bluetooth card, as the screenshots in Step 4 do not match a MacBook Pro 15" Unibody mid-2009 2.53GHz w/ Anti-Glare (I can send you guys a picture of mine if you like).
If you take a look at Step 10 in MacBook Pro 15" Unibody 2.53 GHz Mid 2009 光学ドライブの交換, this is what the inside of my Mac looks like.
But according to Step 4 of this guide (MacBook Pro 15" Unibody 2.53 GHz Mid 2009 Bluetooth Board Replacement), it should like this.
Both guides are apparently for the MacBook Pro 15" Unibody 2.53GHz mid-2009, however both guides have different screenshots. And the guide specifically for the AirPort card doesn't match what the inside of my Mac looks like.
I think something is amiss here. I definitely have a Model No. A1286, "MacBook Pro (15-inch, 2.53GHz, Mid 2009), Serial number: W8934S6N7XJ"--as verified by Apple's website, but my internals do not look like Step 4 above. Instead it looks more like Step 10 in MacBook Pro 15" Unibody 2.53 GHz Mid 2009 光学ドライブの交換.