backup posts and accompanying pics only then restore?

We have a verrryyyyy dated blog at blog.eindividualhealth.com. I am in the process of rebuilding the website, but the blog has me very frustrated. We have Backup Buddy installed but no backups have worked since August. Fortunately, we have only 1 new post since then. Right now, I'm eternally waiting for the Backup Buddy to update itself. Pretty much everything on the site is non-response from the back end. Plugins won't update, etc.
I'd like to dump all the posts with their corresponding vids & pics and rebuild the site, but I need a good workable strategy for that (as well as some technical expertise).
I've attached a screenshot of what I see when trying to update plugins ... nothing, I finally exit out of the page and try again, to no avail.
Do you have any suggestions? I'm afraid anything I do will totally tank the site. Thanks in advance for your help.