Just watch out with tree roots near structures 😁. Or septic fields.
If you can rent/borrow/have a tiller, you can also till in some organic matter or sand into the clay under and around the run to help with drainage in general. But you will probably then need a way to tamp things back down...
You could look into a wood chip trench as suggested that leads to a “rain garden” of water-guzzling plants. If you stick to natives, the should be quite low maintenance. But that would help limit the amount of water heading to the neighbors house.