Yes…and I hesitantly say a migration plugin could be used in place of that, but about the best support you will get for that will likely be little more than a link from someone saying “Read this.” So unless your friend refuses you, server access is definitely preferable, imo.
edit: Another possible option just came to mind…
If your current host allows multiple domains at one hosting account, you could simply add your new domain there as an Add-On Domain and then make some necessary changes to have your new URL land at the existing installation. The overall work would still be similar, but redirecting current links to the new URL would be simpler and you would have access to that in the future whenever the current domain again became used for whatever else.