need to buy logic board

As long as your changing the Logic board why not upgrade to the 2.13GHz board. Mid 2009. they are interchangeable. This will give you a better computer when your done. Make your money count. I would look on Ebay for a logic board. Be sure it is fully tested as working before buying one. As alot of people tend to sell off the old Ones they took out of their computer that were No good. Cost should not be over $225. plenty of guides online as to how to install these. Hope this helps.

The obvious place to start is right here at iFixit:

MacBook Air 1.6 GHz (Original) Logic Board

MacBook Air 1.8 GHz (Original) Logic Board

MacBook Air 1.6 GHz (Late 2008) Logic Board

MacBook Air 1.86 GHz (Mid 2009) Logic Board

MacBook Air 2.13 GHz (Mid 2009) Logic Board

Your serial number confirms that your computer is a Mid-2009 Macbook Air, so you should really stick with logic boards from that generation (1.86 GHz/661-5197, 2.13 GHz/661-5198) for best compatibility.

If the prices for the iFixit boards are out of your range, you can try the online auction site eBay as jeffhogancamp suggests, or use the part number as a search term in the Google search engine to find a retailer closer to you, as Dan and mayer suggest. This method is no different from Googling for anything else; you simply use the part number as the search term.

Parts from iFixit have been tested, and come with a replacement warranty for parts that don't work. Other sources may not check their parts as carefully, or offer any warranty at all. You should check the terms of the sale carefully before you buy. All these logic boards are used, so a warranty is your only real protection against paying for a logic board that's just as dead as the one you're replacing. A warranty is only as good as the seller who gives it to you, so pick your seller wisely.

As for installation guides, the logical one to look at is iFixit's replacement guide.

MacBook Air (Modelle A1237 und A1304) Logic Board tauschen

What's wrong with the old logic board that you need to replace it? If we have more details on the problem, we may be able to suggest less expensive solutions. For example, you can often recover from a liquid spill by washing the logic board in purified alcohol.

Don't do it, all of these are junk and close to death. And there are no newly available MCPs available for them. This is of course assuming that the hinges on your laptop outlive the MCP on the 6 year old board you are about to buy. Take it from my experience this will just end in misery.