If the field is INTEGER, number
you do not quote the value
If the field is TEXT or VARCHAR then value is string
and you need to quote it using '$string'
I do not know your fields. Only you know how CREATE TABLE is.
But here is a guess
$query = mysql_query("update kostenstellen SET
Baustelle= '$baustelle',
kolonnenNr= $kolonnenNr,
Bauleiter= '$bauleiter',
AuftragsNr= $auftragsnr,
Beginn= $beginn',
Ende= $ende,
Gruppe= '$gruppe',
status= '$status'
WHERE kstID= $KSTid;");