SQL Server edition update
Is it possible to upgrade without downtime? As I know maintenance plan
  need to execute in one node. In my case can I execute it on the
  passive node without downtime?


The services will need to be restarted to pick up the new settings. You will also likely be instructed to restart your server. However, upgrading the secondary node and failing over is an option. 


  What is the best backup method for 1 TB database. Backup/ Restore is
  enough for this size?


1TB is large, but large is relative. Following normal backup and restore procedures is the way to go. You want to make sure you are using page verification = CHECKSUM, backup verification &amp; checksum, and if you are worried about the size of the backup then compression.


  Does it necessary to take backup and restore after up-gradation? I
  mean the possibility of data corrupt during the update of edition.


Dozens of things could go wrong. I could caution against an in-place upgrade which is what you are proposing. If something goes wrong, there is no "back button". It would be much wiser to set up new servers, cluster, etc and restore your backup to the new instance after testing is complete.


  Our server has 16 core processor and we have 4 licenses of 4 core
  server (4 x 4 = 16). Should we put only one license copy?


This question depends on what HA/DR you are going to, and what you plan to use the secondary for if you are using AGs. I say this because I assume there is a Enterprise level feature you want to use, or a few of them. However, licensing questions should be discussed with the vendor.