Is it permissible to use an open-source—unencrypted—internet network found in the tenant's apartment, given the interruption of internet service in the tenant's apartment, and knowing that the internet source for all apartments is one?

As long as you are unaware of the source of the unsecured network and that it might belong to another building, it is not permissible for you to use it and benefit from it without the owner's permission. You should ask the landlord to provide the service to you as agreed upon in the contract.