iCloud doesn't sync to iPhone 5s automatically

With iCloud on iPhone 4 (iOS 7), MacBook and Pro (both OS 10.9.3) Contacts, Calendars and Reminders were synced just well.

Now I switched to iPhone5s (iOS 9.1). I didn't install iCloud Drive. Sync works, with one exception: iCloud doesn't sync to iPhone automatically. Changes in Contacts, Calendars and Reminders made on the Book or Pro are shown correctly in the iCloud, but not at iPhone 5s.

I have to connect to iCloud 'by hand', eg. opening Prefs/iCloud, then the iPhone syncs. It also works with Push, but I do not want it activated. Restart, iCloud off-on, etc. only work temporary, and I don't want to upgrade to OS 10.10.

Any Ideas?

Andreas