I had my aortic valve surgery more than 30 years ago. There wasn't a lot that I could have done before leaving back then.
However, if I was to leave today, I would ask the doctors or appropriate staff member if I can have them order an INR test meter. Perhaps insurance cover it if I asked before I was discharged.
You wouldn't be able to use it for at least 90 days while your body adjusted to anticoagulation, but it would be good to have for self testing once you CAN. I'm thinking that it may be better to ask while you're still in the hospital than to try to get a meter under your insurance months after discharge.
OTOH - a lot of people (myself included) have found good deals on eBay. Buying a meter on eBay would save you from some of the hassle of fighting with your insurance company.
Another thing I might ask about is a cushion (or something) that is used to protect your sternum while it's healing. I've seen some postings here about things that help post-operatively.