I’ve needed to fix this twice now. So for my future self who might have tried updating mono because “this is probably fixed by now” and for other people who have the same issue; here is my summary of this topic and other resources to get Sonarr working again.
First, create a back-up of Sonarr (and maybe Radarr?) by copying the contents of /usr/local/nzbdrone/var/.config/NzbDrone to somewhere safe like so:
cp -R /usr/local/nzbdrone/var/.config/NzbDrone ~/sonarr-backup
Uninstall Sonarr, Radarr and mono through the package manager and check if mono is completely removed from the system by running mono -V. Maybe there is an old version still installed and you can remove it manually through synopkg uninstall Mono (v4.x) or synopkg uninstall mono (v5.x).
Download and manually install this old version of mono: https://archive.synology.com/download/Package/spk/Mono/4.6.2-0096/Mono-armv5-4.6.2-0096.spk
I access my synology DS213 via a reverse proxy but for some reason I cannot upload manual packages so connect directly to your nas to upload this package.
After mono is installed, you can go ahead and install Sonarr. This will ask you if you wish to install mono (v5.x) but go ahead and install that.
When you’ve installed Sonarr, Sonarr will not start but don’t worry 
Open /var/packages/nzbdrone/scripts/service-setup in your favorite editor and update the line:
MONO_PATH="/var/packages/mono/target/bin"
to:
MONO_PATH="/var/packages/Mono/target/usr/local/bin"
This will make sure Sonarr will use the older version of mono (v4.x), the difference is the capital M in the package name.
Start Sonarr again, the package manager will tell you that it failed but cat /usr/local/nzbdrone/var/nzbdrone.log will tell you that it’s starting/started 
To restore the back-up; stop Sonarr again (through the ui) and run the following commands:
sudo rm -r /usr/local/nzbdrone/var/.config/NzbDrone/nzbdrone.db*
sudo cp ~/sonarr-backup/nzbdrone.db /usr/local/nzbdrone/var/.config/NzbDrone
sudo chown -R sc-nzbdrone:nzbdrone /usr/local/nzbdrone/var/.config/NzbDrone/nzbdrone.db
That should do the trick so start Sonarr again!