Hi cole,
yes that was it - I got it to get it run first time 
Maybe other people will have a better idea how to build this file, like I didn't have too before, so it is posted here and sensitive information is replaced in the same structure as I wrote it.....
I get now some Active Directory user/password issues, maybe you want to through a look over it?
Unable to verify credentials: LDAP Result Code 49 "Invalid Credentials": 80090308: LdapErr: DSID-0C09042F, comment: AcceptSecurityContext error, data 52e, v2580
I'm trying to fix this too but maybe you will see anything directly 
Thank you again!
My config file:
[Server]
SenderEmail = rstudiouser@domain
Address = https://RConnect.domain
[HTTP]
; RStudio Connect will listen on this network address for HTTP connections.
Listen = :3939
[SQLite]
Dir = /var/lib/rstudio-connect/db
[Server]
DataDir = /var/lib/rstudio-connect
[Authentication]
Provider = ldap
[LDAP "domain" ]
ServerAddress = IP.IP.IP.IP:389
UserObjectClass = Users
UserFilterBase = "CN=DomÀnen-Benutzer,CN=Users,DC=domainDC,DC=local"
UserFirstNameAttribute = GivenName
UserLastNameAttribute = SurName
UserEmailAttribute = UserPrincipalName
UsernameAttribute = SamAccountName
BindDN = rstudiouser
BindPassword = p@s5w0rT
AnonymousBind = false
UserSearchBaseDN = "CN=DomÀnen-Benutzer,CN=Users,DC=domainDC,DC=local"
GroupFilterBase = objectClass={rstudio-user}
GroupNameAttribute = CN
GroupSearchBaseDN = domain
[Password]
WebSudoMode = true
WebSudoModeDuration = 5m
~