Macbook Air logic board upgrade

Solution 1:

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.

Solution 2:

No.

The internal space, connectors, hard drive interface, power supply locations and other internal design changes make this exchange a non-starter.

Even moving a connector 2.0 mm makes a huge difference on a device packed as closely as an Air. The replacement you contemplate is much more radical and would require expensive engineering to retrofit a hybrid device.

Selling the old Air and buying a new device will be far more cost effective than engineering a new custom hybrid device.