Mup works just fine. For cameras that have this feature, the first press of the shutter lifts the mirror, the second fires the shutters so there's plenty of time after the mirror goes up because you're controlling it.
Delayed exposure is OK as long as it delays it long enough (not sure how long that is on the D80).
The mirror does lift when you use live view, of course, but it goes back down and then back up again to take the exposure. However, on long exposures mirror-up isn't that necessary as vibrations settle down before much of the exposure has taken place.