(Interesting 'handle' ...) Anyways, I see your problem. You may go about this a couple of ways.
1) You can build a French Drain that runs to the SURFACE of the landscape right next to the driveway on the neighbor's side of the driveway. Then, when the water runs off it, it runs into the gravel, and down to the drain. Of course, depending how much rain you get, your drain might not hold very much unless you can route the water to another place. If you can, then obviously design the drain so the water is taken to the street or someplace like that. IF you go this route, 'wrap' your gravel and drainpipe in some filter/landscape fabric (NOT 'Weed Block') to about 3" of the surface of the surrounding terrain. Then, cover the remaining 3" with just gravel. This way, if the top 3" of gravel becomes 'clogged' with leaves and pine needles and grass and such, you only have to remove the 3" layer to 'clean off' the contaminates, then re-lay the 3" of gravel you took off.
2) You could 'tilt' your driveway towards the side OPPOSITE that of your neighbor. This way, water can run of the driveway and onto YOUR lot (and leach into the ground, or whatever.) At the same time, you can try to 'catch' the excess water and do as I described above.
Other than that, I'd be sure that your Contractor re-levels your driveway since it's been compacted ONLY where the tires have been riding. My best to ya and hope this helps.