I think this should work with a simple notification. We use the following Code in the SyncAdapter of Mirakel to do so:
mNB = new NotificationCompat.Builder(this.mContext)
.setContentTitle(
"Mirakel: "
+ this.mContext.getText(R.string.finish_sync))
.setContentText(last_message)
.setSmallIcon(android.R.drawable.stat_notify_sync)
.setPriority(NotificationCompat.PRIORITY_LOW)
.setContentIntent(p);
this.mNotificationManager.notify(this.notifyID, mNB.build());
This should be fairly simple to modify for DAVDroid 😉