with regards to this api https://developer.mozilla.org/en-US/docs/Web/API/Cache, I'd like to also set the cache lifetime.
let's say, cache this request for no longer than 10 minutes
That is, unfortunately, not possible.
The doc you linked specifically states that items in the cache objects provided by the Cache API don't automatically get updated nor deleted. You have to implement that sort of logic on your own.
You should note that libraries eg. Workbox already does that for you. You could take advantage of that if other functionality suits your usecase.