Why can't you talk to your neighbor? Seems like the first course of action.
If you insist on taking action without talking to the neighbor, it would be perfectly reasonable to remove the bamboo from you side of the fence. Digging them out is not easy but if you don't they *will* continue to spread throughout your whole garden. Even after being dug out, the will return regularly from the mother plant on the other side of the fence. This should be your neighbor's resposibility but... oh well. Installing a barrier is another possibility... but you really ought to bring this problem to your neighbor's attention.

Digging is difficult work, especially if you're inexperienced. One tip I will give you is that you need to sever the rhizomes in the ground using a root pruning saw or an electric demolition saw with a long blade. This will make it easier to get bamboo out of the ground when you dig it. Once it's under control, you can keep it in check by using a sharp shovel to chop down through any wayward rhizomes coming into your yard. First, however, you need to get rid of what's already there. You're facing a typical case of bamboo that's started to get out of control because no one has maintained it.

As well as the necessity of attacking the rhizomes (only needs to be done at specific times once or twice/year), you have every right to prune the parts of the plant violating your airspace. It won't kill or harm the bamboo. Just Cut It.

I'd use Roundup on any shoots that came up on my side of the boundary.

Herb-
Though I've never tried it, I understand that Roundup is not very effective in taming bamboo. If it were, people wouldn't be having such challenges in reclaiming their yards from runaway bamboo.