If you say that no sooner has one thing happened than another thing happens, you mean that the second thing happens immediately after the first thing.
No sooner had he arrived in Rome than he was kidnapped.
See full dictionary entry for sooner
Examples of 'no sooner' in a sentence
no sooner
Needless to say, no sooner were all these large and rather expensive operations finished than main electricity was brought in and the turbine became obsolete.
Page, Russell The Education of a Gardener (1994)
His experience in dealing with complex family cases meant that no sooner had he retired in 1997 than he was back working as a deputy circuit judge.