Yes, you can do this.
Street Wraith has the ability
Cycling—Pay 2 life.
which (as the reminder text states) is shorthand702.28a for
Pay 2 life, Discard this card: Draw a card.
Since this is an activated ability, everything to the left of the colon (:) character is the cost that must be paid to use the ability.602.1a You pay the cost (discard the card and pay 2 life) and the ability is placed on the stack.
Once the ability hits the stack, it follows all the usual rules for resolution - including allowing players to respond by using different spells or abilities.602.2b At this point you can use Noxious Revival; it also goes on the stack.
Once everybody passes priority, the spells resolve in LIFO order: Noxious Revival resolves first, letting you place the Street Wraith on top of your library, followed by Street Wraith's cycling ability, which lets you draw a card.