You can run Memberium on GoDaddy Managed WordPress, but it would not by my first recommendation, and I would use it with caution.
Managed WordPress has a pretty heavy caching layer that works at odds with using it for a membership site The goal of the caching layer is to capture the web page and deliver that same page over and over to every visitor for maximum speed and efficiency, which is great for something like a sales and marketing website. There is no way I know of to tune or disable this caching.
For a membership site, the goal of the software is to deliver a personalized and customized experience for each member. Obviously you don't want Johnny to get access to Jane's membership material, or to get Jane's "Access Denied" screen just because it was cached.
The above may be a customer service inconvenience issue, but it can be even more serious if Johnny goes into his "my account" page and sees Jane's address and credit card info.
These issues are issues of concern when using any heavy page caching on a membership site, and aren't unique to Memberium or to GoDaddy's Managed WordPress product.
Disclaimer: I am the founder, and author of Memberium.