Yes you do not have to do anything, just wait that the domain name is translated to the new IP of the server instead of the old one, thanks to the DNS servers.
In case you (or anyone using the same ISP) did not access the site in the few hours before the change, it was not in their cache, so you do not have have to wait at all. Same if your ISP does not cache the DNS entries for a long time.
Some others may cache that for 24 (official delay) or even 48h.
A little before doing the switch I changed the delay from 24 to 1h, if I had done that yesterday it would be even more seamless for everyone