In order to play Anno 1404 on Proton, follow these steps.

Install DirectX

WINEPREFIX=/home/<user>/.local/share/Steam/steamapps/compatdata/33350/pfx
wine /home/<user>/.local/share/Steam/steamapps/common/Anno\ 1404/DirectX/DXSETUP.exe

Edit Engine.ini file

The content of the file

/home/<user>/.local/share/Steam/steamapps/common/Anno\ 1404/Engine.ini
<InitFile>
<DirectXVersion>9</DirectXVersion>
<UbiSurveyTime>-1</UbiSurveyTime>
<UbiSurveyTimeStatus>2</UbiSurveyTimeStatus>
</InitFile>