How does the respiratory system of a praying-mantis work?

They do not breath through their nose- in fact, they don't even have a nose!

Praying mantises (and all other insects) breathe through spiracles. Spiracles are small holes in the sides of the insect which allow air inside, and the oxygen diffuses directly into their cells.

They do not have lungs. No insects have lungs-- Spiders do have something that somewhat resembles a lung (a book lung) but it's not really a lung.

Insects and mammals evolved almost completely separately, they don't share a lot of common features structurally.