I suppose there's a launch window to Neptune by using Jupiter as often as there are Jupiter/Neptune conjunctions, every 13½ years. But which years are Jupiter and Neptune best aligned for gravity assist? How long does such a launch window last (three months, to take advantage of Earth's orbital speed?)
Using the Trajectory Browser, I was able to figure out the following trajectories that are available.
The dates are Jan-18-2018, Feb-22-2019, Feb-23-2031, and Mar-29-2032.
So basically there are two good launch windows roughly every 13 years To show out the path a bit more, I've included more than just the optimal paths.
Unfortunately the tool doesn't show how long the optimal window lasts, but it can be inferred that it lasts for some time. The total mission time is about 10 years for such a mission, depending on exactly how much one wishes to optimize the fuel expenditure.