Learn Russian: Не успел он открыть зонт, как ударила молния. - He had no sooner opened the umbrella than lightning struck.

In the fixed sequence construction не успел ... как ... (no sooner had... than...), the second clause is always introduced by как. A literal translation is closer to "He didn't have time to open the umbrella, when lightning struck." You cannot use чем here because this is an expression of time, not a direct comparison.

Why do we use the perfective infinitive открыть after не успел?

The verb успеть (to have time / manage to do something) almost always takes a perfective infinitive. This is because the focus is on achieving the final result of the action (getting the umbrella entirely open) before the sudden interruption happened.

Why is the word order ударила молния (verb-subject) instead of молния ударила?

Placing the verb before the subject is a common way in Russian to introduce a sudden new event, especially with natural phenomena. This verb-first word order emphasizes the suddenness of the strike, which perfectly matches the dramatic "no sooner had..." mood of the sentence.

Is зонт in the nominative or accusative case here?

It is in the accusative case because it is the direct object of открыть. Since зонт (umbrella) is a masculine inanimate noun, its accusative form looks exactly identical to its nominative form.