Get an external hard drive, 2.5 to 3 times larger than the storage capacity of the replacement drive you will install.
Plug it in, click yes when the finder asks if you want to use it as a time machine drive.
let it backup the entire drive. when that is done and confirmed backup, then eject the drive and unplug it when safe to do so.
Then get an 8gb usb flash drive, and follow the instructions on making a usb installer disk for MacOS here :
https://www.lifehacker.com.au/2016/09/how-to-make-a-bootable-macos-sierra-usb-flash-drive/
Then eject the drive when done, and unplug when safe to do so.
Install your new drive. When you have it in and all ready to put the operating system on, do this :
Plug in the 8GB install disk, then turn on the Macbook Air while holding down the R key.
You will see that it shows the disks, click on the install MacOS USB disk.
Install MacOS, it will take around 30 minutes to do and may restart during the install.
Then at the setup of the new MacOS system screen, it will ask you about using a time machine backup to transfer from an old mac. (This should be the Migration Assistant app if you skip it and do it after the system is started fresh.)
You then plug in your time machine external hard drive, and select the time machine backup.
You let it copy back over all your old apps, documents and settings, etc.
At the end of that, you have your old stuff back on the new drive, with the extra space and a time machine backup disk the right size for your new drive, an installer USB disk which you can reuse.
You may then want to get a USB enclosure for your old 128GB SSD, which lets you use the old mac drive as a large USB 3 flash drive at high speed.
https://eshop.macsales.com/item/OWC/MAU3ENPRPCI/
Either way, it seems that the larger SSD should go slightly faster than the 128GB original SSD due to increased capacity differences. Enjoy !