I think you're out of luck. From iFixit, here's the logic board from the original MacBook Air:

And here's the logic board from a 2012 MacBook Air:

Unlike desktops, which typically use standardized parts, laptops use fairly specialized components, and on top of that, Apple tends to iterate its designs fairly aggressively even within models with similar outwards appearance to achieve better cost or performance (e.g. smaller logic boards means more space for the battery). Because of this, most Apple components beyond hard drives and RAM are only interchangeable with those of the exact same generation.

I assume you're looking for a way to bypass the power button to power it on? If so this should help.

The power pads, two slightly raised solder points, are found just above the keyboard flex connection on the logic board. See the circled pads in the picture below. Bridging these two pads while power is provided to the Mac should power it on. Be sure to use a metal tool with a non conductive handle to bridge the two points.

Source picture can be found in iFixit's 13" Mid-2011 MacBook Air logic board replacement guide, used and modified under a CC license.

Something to keep in mind, a stuck power button will prevent a Mac from powering on even when the power pads are shorted. To verify that the top case (it houses the power button) is at fault you should disconnect the top case flex cable (silver flex in the photo) and try again if the Mac doesn't first power on with the power pads. If it still doesn't power on with the flex disconnected then the logic board is likely at fault although you could still dive deeper and disconnect components (SSD, speakers, display) to further isolate the logic board.

Good luck! Hopefully it's something simple/cheap.