apt-get update quits without message
Is there any way I can update the server without having to resize it?
Yes, add a swap partition to help cope with the temporary increase in memory usage.
sudo -s
touch /swapfile
fallocate --length 4G /swapfile
mkswap /swapfile
swapon /swapfile
chmod 600 /swapfile
You should now see that there is an extra 4GB of swap available. This won't persist after reboot, if you want to do that you'll have to add a line to your /etc/fstab like this:
/swapfile none swap sw 0 0
It really needs 2+ GB RAM to function at its best