If his lawn isn't draining properly then that's probably his problem. Gravity can be a bitch sometimes. The only way that I can see that you have any liability is if you have visible runoff from your lawn - ie your lawn isn't draining downward as much as it should.

Even if it is your problem, he has to have given you a _reasonable_ chance to fix it before suing you. Telling you that he's going to get the builder to do something isn't doing that. Nor is threatening you by proxy when you are at work.

First step should probably be to get your city inspector out to look at the problem. They should be able to advise you on this.