What was the extent of Kohl's government opposition to the independence of Ukraine?

There's actually an English version (i.e. Der Spiegel International translation) of that article, but it has a somewhat different title and doesn't mention Ukraine in the summary, so it's a bit hard to find/relate. It refers to the same print edition 18/2022, that's how I know it's the same piece referenced.

The article is mostly paraphrasing with only the briefest of quotes from actual/original documents. But there there are two primary sources quoted there in re Ukraine: one is Kohl's conversation with Mitterrand "in early 1991" and then 2nd a November memo summarizing a discussion between Kohl and Yeltsin, a couple of weeks before Ukraine's independence referendum.

The new [declassified] volume with papers from 1991 includes memos, minutes and letters containing previously unknown details about NATO’s eastward expansion, the collapse of the Soviet Union and the independence of Ukraine. [...]

In 1991, the Soviet Union was still in existence, though many of the nationalities that formed the union had begun standing up to Moscow. Kohl, though, felt that a dissolution of the Soviet Union would be a "catastrophe" and anyone pushing for such a result was an "ass." In consequence, he repeatedly sought to drum up momentum in the West against independence for Ukraine and the Baltic states.

[...] Kohl found himself faced with the three Baltic republics pushing for independence and seeking to leave the Soviet Union, Kohl felt they were on the "wrong path," as he told French President François Mitterrand during a meeting in Paris in early 1991. Kohl, of course, had rapidly moved ahead with Germany’s reunification. But he felt that Latvia, Estonia and Lithuania should be more patient about their freedom – and should wait around another 10 years, the chancellor seemed to think at the time. And even then, Kohl felt the three countries should be neutral ("Finnish status"), and not become members of NATO or the European Community (EC).

He felt Ukraine should also remain in the Soviet Union, at least initially, so as not endanger its continued existence. Once it became clear that the Soviet Union was facing dissolution, the Germans were in favor of Kyiv joining a confederation with Russia and other former Soviet republics. In November 1991, Kohl offered Russian President Boris Yeltsin to "exert influence on the Ukrainian leadership" to join such a union, according to a memo from a discussion held between Kohl and Yeltsin during a trip by the Russian president to the German capital of Bonn. German diplomats felt that Kyiv was demonstrating a "tendency toward authoritarian-nationalist excesses."

When over 90 percent of Ukrainian voters cast their ballots in favor of independence in a referendum held two weeks later, though, both Kohl and Genscher changed course. Germany was the first EC member state to recognized Ukraine’s independence.

Three's a bit more about NATO expansion in there, but since my Q wasn't about that, I won't get to the remained of that article.

Judging from the text of the article you linked - Kohl was expecting the dissolution of USSR to result in a surge in toxic nationalism not unlike the aftermath of WW1. As we can see from history, the dissolution of a multi-national state allows the tensions between its peoples that were suppressed by the empire's law enforcement to grow into full-fledged wars - and indeed fall of USSR did lead to multiple conflicts between or within post-USSR states (Karabakh, Chechnya, Tajikistan to provide some examples). Kohl was understandably not eager to see a war brewing so close to Germany's borders (and what's more, a war concerning countries important to German economy). Thus, he attempted to somewhat soften the transitional period - instead of a collection of completely independent states, he envisioned a confederation that would allow the constituent countries of the Union to "ease into" independence. Allowing some countries (Baltic states, to be precise) to leave wuld create a domino effect, Kohl argued, according to the article (as translated by Google) -

If the Balts left the Soviet Union, the Ukrainians would follow and eventually the entire Soviet empire would perish, including Gorbachev. That's pretty much what happened in the course of 1991. Only Kohl doubted that such a dissolution would end peacefully. He considered a kind of 'civil war' possible, as would soon be the case in Yugoslavia.

Same train of thought was applied to NATO membership (again, via Google translate):

In February 1991, Bonn's ambassador reported that the East-Central Europeans' urge to join the defense alliance was creating an explosive mixture of "perception of a threat, fear of isolation and frustration at the ingratitude of the former brother countries".

Genscher [Kohl's vice-chancellor] did not want to fuel this situation any further. NATO membership for East Central Europeans was "not in our interest," he declared. Although these countries have the right to belong to the defense alliance, "the point now is not to exercise this right".

The Spiegel article seems to only operate with documents concerning 1991, so it isn't clear how Kohl's policy changed when it was clear the USSR was done for concerning independence of post-Soviet states; but we do know that NATO enlargement into Eastern Europe only started after Kohl left office in 1998.