What information is meant to be conveyed by the sentence?
It would be more common to say 'in a rush', meaning in a hurried way. Unless you tell us the intention of the speaker or writer, we will not know if that is what is meant. A rush is also a plant. I can imagine contexts in which that definition of rush might fit. It's not nearly so common, but I wouldn't exclude it without some context.
To me, "He does things with a rush.", sounds like the speaker does not know that the common idiom is in a rush.
Of course it is possible that there are regional variations and someone says "with a rush" where I would say "in a rush".
A rush is a kind of grassy plant that is sometimes used to make woven baskets and things like that. Who knows what he could do with a rush.
A rush, is a term used to describe the adrenaline jolt we get when we do something exciting or dangerous. Typical use, "That rollercoster ride is great, what a rush."
Maybe he is addicted to risk taking and does things with a rush.