upgrading logicboard from 1.86 GHz 2GB to a 4GB

Sure you can swap out your logic board with 4 GB of RAM

• 13” MacBook Air Logic board, 1.86GHz, 4GB: Apple P/N 661-5798 - Aprox $180 USD
• 13” MacBook Air Logic board, 2.13GHz, 4GB: Apple P/N 661-5734 - Aprox $200 USD

Here’s the guide MacBook Air 13" Late 2010 Logic Board Replacement

The OS is on the SSD drive which you’ll transfer to the replacement logic board so you won’t loose anything or need to re-install your OS.

But with what I’ve said here I don’t think this makes much sense given the age of the system and the cost as well as availability of the part. I would strongly recommend looking for a newer used MacBook Air or Pro system as being a smarter investment.