This blogpost will shortly describes the steps what I did when I did the ASR between my on-premises (VMM/Hyper-V) and the Microsoft Azure from DNS settings point of view. Reason why I did the following changes was that I want to update DNS records automatically and fast after the failover.
First I changed the TTL time from 20min to 5min on that VM which I´m going to use with ASR Failover.
I also Allowed dynamic updates.
Open DNS Manager.
In the console tree, right-click the applicable zone, and then click Properties.
On the General tab, verify that the zone type is either Primary or Active Directory-integrated.
In Dynamic Updates, click Nonsecure and secure.
The zone must be enabled for secure and non-secure updates.
With this way the correct IP address is updated to the DNS after planned failover.