Need advice to prevent runoff water from going into neighbor's yard
..Your driveway would be the cracked concrete slab to the left, I assume?

If the picture is not misleading, there appears to be some slope towards the street. If so, rent a concrete saw or buy a concrete blade for a saw you own (but unless you own a water-cooled saw, a rental saw that is water-cooled will generally do a better job, faster) and cut a bunch of grooves 1/2" (or so) deep leading to the curb/gutter at streetside.

That may be all you need to do, or you may need to make some sort of raised edging at the right side of the driveway. That would probably best suit the area if formed from concrete, but something like a rubber glue-down garage door bottom seal (the type that's stuck to the floor, not the type that's stuck to the door) might be quicker and cheaper, until it dies in the sun and needs to be replaced.