Premium hosting has generally higher limits, and many of the limits that are fixed on free hosting can be configured on premium hosting.
But I cannot provide any guarantee that any particular plugin, feature or functionality will work correctly on premium hosting. Ultimately, we just provide server space, after all. And you haven’t even said which backup plugin you are using (there are MANY different ones).
If being able to use a backup plugin is reason for you to use premium hosting, then you do you. But if all you want to do is make a backup, there are perfectly fine ways to do that on free hosting. Just not with backup plugins.