Should I enable binlog binary logging on slave server?
Enabling binary-logs on your current slave server will eliminate two steps of the failover process but there would still be many other steps involved. 

In a true failover and depending on the scenario you'd probably go through the following steps.


Enable binary logs on the new master via my.cnf or equivalent.
Restart MySQL on the new master.
Grant the old master's replication user access to the
new master.
Run the FLUSH TABLES WITH READ LOCK command on the new master.
BACKUP the databases being replicated on the new master.
Run the UNLOCK TABLES command on the new master.
RESTORE those databases to the new slave.
Run the stop slave command on the current slave.
Run the CHANGE MASTER command on the old master to point to the new
master.
Run START SLAVE on the new slave.


Update after Karoly's comments

As a follow response up to Karoly's answer and comments. If you really wanted to save time on the fail-over process, using a master-master replication schema would be the best idea in my opinion. 

You would essentially prepare the slave to take over as master at anytime, leaving you with just redirecting your applications to the new master. 

Assuming you've already setup a master-slave replication schema. 
The Basic steps to then setup a master-master replication, you would.


Turn on bin-logs on the slave
Turn on log-slave-updates on both the master and the slave
Configure the master to update from the slave with CHANGE MASTER
Grant access to the replication user on both master and slave
start slave on both servers
Eventually make the slave server read-only via the read-only my.cnf
option