How many solar cycles to make our solar system?

It could have happened in as little as one cycle, if the star in question was a high m star and exploded as a supernova. Supernova are the only things powerful enough to naturally create elements heavier than iron, as iron fusion is endothermic -- you need to add energy to make it work, energy that is in abundance in a supernova. Our birth nebula COULD have resulted from a starburst due to a small irregular galaxy ping through the Milky Way's disk, but because of our tidy, only-slightly-elliptical orbit about the galactic center, I'd guess the pre-solar nebula's collapse was entirely due to happenstance within the Milky Way, and not due to an interloper.