Standby Master Server

A clustered master is the best to satisfy your requirement, but that needs to be done upfront. 
An existing standalone master server can only be converted to a cluster as consulting engagement.

A standby Linux server with same hostname and different IP address that is not on DNS can probably work.

The issue is not the hostname - that is easy to update/change.
The issue is keeping the catalog in sync with the production server.

Catalog replication is one way, but you need to ensure that ALL catalogs and db's reside on the same lun and that write-order fidelity is guaranteed by the replication technology.

This document explains the options in detail: NetBackup in Highly Available Environments Administrator's Guide 

 Have a look at the section under the topic: "About non-clustered NetBackup master server with catalog replication".

PS:
Your current environment is at risk as NBU 7.6 is no longer supported.
Please plan and schedule upgrade as a matter of urgency.