I got a notification to update via Software Updater however it says I don’t have enough free space available. I have 1.7GB free on the sd-card however it still says I need 300 or so more MB when it gets to the packages installation step and then aborts.
Also, would I have the same amount of space free (1.7GB) after updating or would it leave me with no space at all ?
Upgrading the distribution on the Raspberry Pi isn't really recommended, since it's not supported, could take a really long time, or yes, it could run out of space.
It's likely to miss out the new kernel too, which is upgraded separately:
sudo rpi-update
A fresh install ("flash" / restore to the SD card) is recommended where possible.
If you'd like to upgrade anyway, you'll need plenty of space. Running out during installation will likely bork the system. I recommend installing BleachBit from the Software Boutique to clear out junk files.
After an upgrade, there would be cached packages, which can be cleared using BleachBit or the terminal: