If I understand this correctly
and if you neighbor doesn't eradicate it 100% by removing the bamboo you may have an out-of-control problem.
Bamboo is a rhizome/grass and grows in culms which spread out underground and are just invasive as hell.
http://en.wikipedia.org/wiki/Rhizome
Your question: "Will the roundup work?" Yes, it just might. It also might ruin your soil for 2+ years from what I've read. You will have to locate every single culm. (extremely difficult)
It's all or nothing. And the neighbor MUST kill his as well or it simply will not work to prune or "cut it back" off of your property.
How large is this problem? Is this the common golden bamboo?
It sounds like your problem is minor right now. Thats the good news.
Personally...if it were me: I would hire some labor to shovel and pick ALL Culms out. Having a pro look at it is best.
If its near your home, driveway etc then it's very important that you do not ignore this. Culms start growing underground and you wont even see it until it pops up one day.
Good move in seeking advice now...