Is this the right use of the word 'lest' in this sentence?

Solution 1:

Both forms of the sentence work and are understandable. However, the should version is expressing an unlikely situation. That means that it is wrong to be early or on time and that something, somehow is forcing him to not be late.

The shorter, without should version is the likely and ordinary situation. It means that he should not be late and he is worried that he will be late.

Solution 2:

Both versions mean exactly the same thing: "He was worried that he might be late."

The second version uses the subjunctive form of the verb "to be," expressing uncertainty or a deviation from reality.

But the subjunctive form of the verb looks identical to the infinitive. So in the first sentence, a modal verb has been added to clarify that it is the subjunctive. (I've actually done the exact same thing in my example sentence above.)

This is a common technique in speech and in writing that is less than extremely formal. "Should" and "would" are commonly used for this, since they're basically permanently in the subjunctive anyway.