I'm a tenant, is it my responsibility to manage my services during the PSTN switch off or my landlord's?

It depends on who the phone/broadband contract is with. If the account is in your name, it's on you to engage with or respond to your provider. If it's a service the landlord manages (such as a shared entry system or building alarm), that's the landlord's responsibility to sort out. It's worth telling your landlord about anything in your flat that uses the phone line, just in case.