Well, if the planet were orbiting a typical (main sequence) star, then I would imagine that the stellar winds would clear the local astrosphere of any gas or dust from the nebula. So there wouldn't be necessarily any more shooting stars than what we see here on Terra. Depending on the activity level of the star, however, there might be visible interplay in the form of bow shock between the nebula and the edge of the astrosphere; I imagine that might look something like a brighter and more diffuse version of zodaical light. Or perhaps like a subtle aurora that never noticeably moves, except in timelapse photography.
On the other hand, a planet in orbit around a star that was in the process of making its own nebula (like an AGB star or a very recently formed planetary nebula) would be bombarded with stellar debris on a continual basis, and as long as the planet still had some sort of atmosphere and/or magnetic field to interact with, you ought to be treated to some rather spectacular and regular aurorae and/or shooting stars.