FYI, a 120GB SSD is barely large enough to run macOS before you even store any personal files on it. That is the sad truth here. These days it is important to have 80GB-100GB of Free storage space before starting any workloads since that 100GB of Free space can disappear very quickly. Older versions of macOS could get away with just a 120GB SSD, but it was always a very limited in space.
Many old MBAirs also had just 4GB of memory with 8GB of memory being the most available.....the latter is a problem, but the former is a huge hindrance and can easily cause more storage space to be consumed due to Virtual Memory (aka "Swap").
Now combine that with the macOS 12.x Monterey installer being very large due to it being a Universal app containing both Intel & Apple Silicon executable files (later versions of macOS have even larger installers). Apple's documentation mentions the Monterey installer requires at least 36GB of Free storage space when running the installer while booted into macOS (45GB if using Recovery Mode or a bootable macOS USB installer). I think those are the correct values, but I can no longer find the Apple article & I'm having trouble locating my older posts from four years ago.
Yet another complication is the very misleading "Available" storage space value shown everywhere in macOS which is not synonymous with Free. The only place you can actually see the actual Free storage space value is in Disk Utility on the right side just below the storage bar graph.
It is very hard to create Free space on SSDs which are so small. We see the same problem even with systems with larger 250GB SSDs. Once the boot drive is very low on Free storage space it is really hard to recover that storage space.
You can try booting into Safe Mode to see if that makes any difference since fewer apps & processes are running.
Before you attempt to upgrade macOS you absolutely must make a good backup of your computer to safe guard your important files just in case something goes wrong.