New Shimano wide range cassette 8 speed 11-40 HG400-8

The thing is that you won't be able to serve this cassette by an old rear derailleur. Those had been for 32 or 34T biggest cog at max.
With a long b-screw and short chain (1×X) you could get very likely 36T still work.
For 40 and bigger you need a goat link, but this has negative impact on precision on small cogs.
Ideally simply use a proper almost brand new rear derailleur designed for it... Sure, those look all awful on a classic bike.
 
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