"Is there any issues in calling Unmanaged exe in managed exe"
No - when you start an application with Process.Start it's a completely independant app which shares nothing with the app that started it. It can be anything, including a file that causes Windows to open the associated program such as a web browser. Managaed / unmanaged is irrelevant.
But...The code you show won't compile as the FileName you set isn't a string.
Assuming you mean:
s.FileName = "example.exe";
instead, then the most likely reason is that Windows can't find the file you specified. If it isn't in the current directory, or in a Windows application folder, or on the Windows Path then it won;t be found and that might cause your problem.
I'd suggest trying by giving an absolute reference to the EXE file:
s.FileName = @"C:\Program Files\My Examples\example.exe";
And see if that fixes it.
Alternatively, if the file is being found by still shuts down immediately, it's probably that it needs some parameters you aren't supplying - so start by executing the app via a command prompt: Hold the Windows Key and press "R", then type CMD and press ENTER. Type the app name and see what happens. Probably, you'll get a "File not found" error, so use CD to get to the right folder, and try again.