Don't cap the pipe, its not on your property so it would be trespassing, tampering, and who knows what else if you cap it, the water backs up, floods his basement, etc.....
Take a look at the terrain and where you can direct this water to...can you bury a drainage pipe and grate at the outlet of the pipe on your side to catch this water and direct it to the stream?
At our previous house, there was a wet spot right between the neighbors house and mine, and general drainage issues along that properly line...we worked together and dug a trench, jackhammered out a huge rock that was buried in between the houses, put in a drainage pipe and some grates to catch the runoff and send it to the storm sewer, it worked out great....we even hooked our downspouts and his foundation drains to this drain pipe, it really dried up that swampy area.