No. The original sentence is correct if you change "then" to "than". Sentences that start with an adverbial phrase, often require a subject auxiliary verb inversion.
Student or Learner
Hi.I saw the following sentence in a web site and I didn't understand why 'had' was used before subject.Can you please explain it?
No sooner had I started to swim then it began to rain.
I know that auxiliary verb comes after subject so this sentence must be 'No sooner I had started to swim then it began to rain'
Is my sentence correct?
You could also say:
As soon as I started to swim it began to rain.