Unusual usage of 'no sooner ... than ...'

In this case, 'sooner' is being used like 'rather'. We can use 'would sooner' to talk about a preference, usually of one thing over another, e.g. I would sooner go hungry than eat eggs. We can also use 'would no sooner' to talk about unwillingness to do either one of two things; often (but not always) the second thing is unrealistic or ridiculous and used to emphasise the dislike of the first option - I would no sooner go out without a hat than I would go out naked; I would no sooner think of writing something out first of all than fly to the moon; I would no sooner choose Elementary over Sherlock, than I would eat my own hair.

Sooner