you normally can't assign an ip address two 2 machines. so don't. you will just have to cut over the ip address cold turkey when you are ready.
The only time two servers can "share" the same IP is if you're using Clustering/Network Load Balancing. However, each server individually needs their own IP, and accept incoming connections on the Cluster IP.
It's best to assign a new IP to the new machine, and do the cut over when ready.