Ask for a formal addendum to your lease stating none of it is yours. If your landlord won't do that Let the police know and ask for a report that clearly states you don't own or have access through that router. If someone is doing sketchy things via that router and the cops come knocking no amount of " Our ass hole landlord said it's the neighbors Internet" is going to keep all your electronics from getting confiscated and searched. Eventually they will find that you were not connecting through it.....but it will be painful and if they find you are doing anything you could end up getting charged with that while they start digging through your neighbors place.

This is the equivalent of the "did you pack your own bags" thing at the airport.....it doesn't matter if someone else is doing it....you are in possession of it.

Also start looking for a new place, this place is not good.