I was able to get around this.
First I looked up VS in the start menu and clicked "Open File Location" which lead me to a shortcut to the school server. Then I made a link to that link on my desktop.
In Unity went to preferences and set the default editor to that like. Then I opened a .cs file and it generated the solution.
On my machine it opened ALL of ...
Well you can create a solution from a console with the dotnet SDK:
dotnet new sln
And add Projects with:
dotnet sln <SLN_FILE> add <PROJECT_PATH>
I don't know if Unity works with a default C#-Solution, or if it needs a.
(Of course you need to replace the <SLN_FILE> and <PROJECT_PATH>).