What dose that means to the node?
Nothing. Zombie nodes issue was fixed a long time ago. Your node is fully operational if it is connected to the Internet and up-to-date with the Bitcoin blockchain. You don't have to change anything in your node's settings. Zombie node referred to a node which was offline and still had been considered capable of routing payments.
What happens if I'm offline will the sender even be able to send? Since it has to be online all the time can I run a full node on my phone which is pretty much on all the time?
The payment will fail after some time if not immediately. Running a node on your phone would quickly drain your battery.