Typically when you are setting up an Active/Passive cluster, the only software license cost you have is for the Windows OS. So for example if it is a SQL Server, you only need to license one copy of SQL Server.
When patching is done correctly both nodes of the cluster shouldn’t ever go down at the same time as you should patch each one separately and not patch another node until the first node comes back online.
An additional point to factor in is the loss of productivity while everyone works to get one server restored to working condition verses having the other server take over and losing no productivity while the first server is brought back to service. That is what typically ultimately decides whether it is worth it.