Hunter did commit the crime (as the statute is currently interpreted by federal courts), but there is a strong argument the gun possession law is unconstitutional. The reporting crimes are far less vulnerable on Second Amendment grounds, however. That conviction is more likely to stand.
Sullum is also right that this particular crime (possessing a gun while using or being addicted to illegal drugs) is very rarely prosecuted. Tens of millions of Americans are guilty of it, and the feds only go after a tiny fraction of them. Hunter Biden got nailed in large part because of his prominence as the president's son.
I would add that I believe the federal drug laws at issue here are also unconstitutional, because beyond scope of Congress' authority. Sadly, awful Supreme Court precedent (Gonzales v. Raich) says otherwise. My views on both guns and drugs are longstanding, and not driven by this case.
I do recognize, of course, that Joe Biden is a big supporter of the underlying gun laws here, and many of the drug laws, too.
But the sins of the father should not be visited on the son. By the same token, the son's bad behavior should not be held against the father - especially when the father is running against a far worse criminal and authoritarian.
That said, I also think Hunter Biden is a grifter and an awful person. He may well deserve punishment for the tax law violations he's going on trial for later this year. He's not as bad as Trump (and, of course, not running for president). But you can be less bad than Trump, yet still be pretty terrible.