Again, thanks for your answers,
maybe someone can replay these steps:
-Turn off mobile data
-Connect to Wifi
-Put phone in standy by, and don’t wake it!
-Leave Wifi range for some minutes
-Get back into wifi range, but don’t wake the phone
-Write your self an email from somewhere else, eg an PC
-Wait for the to be received on the phone, but don’t wake the phone
-My result: the email is not received, no vibration, no sound, no LED notification
-Wake up the phone, Wifi is not connected, after some seconds the Wifi is reconnected, and the email is received
Since my phone has already broken several times during the update, I am currently hesitant with updates