I Was in Falmouth in Sep. and going on Voyager Feb. 11. It appears that Montego Bay is about 30 min. cab ride. There is a shore tour that does go there. The port is brand new and has all the shopping (Diamonds International, Colombian Emeralds, etc.). Leaving the port is a bit intimidating as the locals set up a straw market and gather at the entrance to try and sell stuff, give rides, etc. But I walked out on my own and went around to take pictures of the ship. I found myself alone in the residential district. The few people walking were very nice. But I will admit that walking alone is probably not a great idea. However according to web comments Falmouth is safe and quite historic. There are some recommended eating establishments and bars in town and of course the standard Jamaican markets (do a web search). We plan to simply stay in the port area and hang out and enjoy the ship. But there are beaches etc. nearby (cab rides away) and supposedly the cab fares are fixed. This port is a HUGE economic boost for Falmouth so I expect the locals will treat passengers very well to promote their income.
Hope this helps.