The best way is to hire a developer who can actually do the coding part and upgrade the obsolete code from the software. He/She might charge a premium, but this is the only way to go ahead.
Or else, you might have to take a backup and then post...
If you know to code or can learn, then you can hand-craft a fix to every depreciated code that there is. You don't have to be careful with new updates as the Theme Developer is not going to update the theme. Even if he does, just ensure that you...