Shouldn't I just be able to go to mysiteaddress:4201 and the studio should come up?

yes, but it depends, if you try to access the server from another machine different than the server using the mysiteaddress:4201 then the problem can be somewhere else and related to dns or ip resolution, have you tried opening the page in the same server with localhost:4201 or 127.0.0.1:4201? if that works but mysiteaddress:4201 not then it means that you have something not configured right in your network level.