Hello and welcome to the forum! You may want to provide further information on the OS X installed on your MBA and its exact model - appears that your current BT version is 2.x (HERE) - BT standards changes usually include hardware, firmware, and software upgrades, so changing the BT radio in your laptop would be unlikely (probably soldered into the MB, but I'm not one of the hardware experts here on older Macs).
However, you might want to consider a BT dongle that plugs into a USB port - plenty available on Amazon HERE - I've never used one so cannot comment on their compatibility w/ older Apple hardware and OS X - hopefully, others will 'chime in' w/ their own experiences. Good luck - Dave
Hi again - as stated at the end of my previous post, I have no experience in using a BT USB dongle on an older laptop - also, I'm not sure that if you first disable (i.e. turn off) BT on the laptop and then plug in the BT dongle, whether the device would work? Likely there are drivers installed and the device might function?
Take a look at this Amazon LINK to an Iogear BT dongle - review the comments carefully - there seems to me a mix of positive and negative statements regarding what you would like to do - the devices are at least cheap -
Hopefully, a member or two w/ experience in this area will come along to help solve (or discourage?) your issue - also, you might want to tell us why you need to go beyond the BT 2.x standard now on your laptop, i.e. is BT not working and/or do you want a faster connection, and if so, for what purpose. Dave