How can I prevent the water from my neighbor's leaky gutter from entering my basement?
I guess the 'repair' solution for the neighbor's house didn't pan out.

You can use your friend's solution, you are just limited by how much room you have to the property line.  You just excavate to the frost line, attach an waterproof isolation layer to the house and extend it above the new pad.  Place some curbing forms and pour concrete.

Another method (even more expensive) is an exterior french drain with a full wall membrane

The interior french drain/sump pump combo will also work.  Unfortunately, you'll need to trench all 4 exterior walls with this method.

Recently, others on the site have used wall coatings (Dry-loc sealer) to stop small amounts of water intrusion.  YMMV.