Sometimes, you have a system you just can't update as quickly as you'd like. Maybe the system can never be taken down. Or it has to run an old operating system because of a specific software package.
These things happen.
But you still need to make sure these systems can't be compromised. After all, if it can't be taken down to be patched, you really don't want it to come down at all, right? and that includes being taken down by malware.
So, if you can't harden the system, you need to harden its surroundings. You'll need to depend on perimeter-based defenses. You'll need to isolate the system, monitor the environment, and track incoming and outgoing traffic. And you can't put software on the system to do it.
To start with, isolate the system. Put it behind a simple switch, and lock the switch down. You don't need to spend a ton of money on this either. A small, simple switch can be configured to only allow configuration from physically attached systems, for example, and you can then configure the switch by physically plugging in a laptop and using the ubiquitous web configuration interface. Only allow the traffic that must pass through the switch, nothing else.
You'll need to monitor the environment too, and the best way to do this without touching the read-only system is to track incoming and outgoing traffic. A way to do this is to configure one of the ports on your switch as a port mirror so that you can capture all the traffic coming into the switch for the read-only system and all the traffic sent by that system to the wider internet. This can end up being a large amount of information, potentially, so you'll need to decide how long you can maintain that information. This, of course, does require another PC, but that PC doesn't need to be expensive, and it can actually run a more modern operating system. You'll need to remove captured log files on a regular basis, and you'll likely need to spend some time analyzing the captured data for possible anomalies.
We have, of course, added new systems to secure the read-only box, but this results in a negligent increase in the overall attack surface. The new monitoring system should not expose services through the switch, although data will still need to be removed. This might be addressed either by USB-based data removal via sneakernet or a secondary NIC and a control network. Both of these possible configurations will require appropriate security controls, of course. And both the monitoring PC and the switch should run the most recent and patched operating systems available.
Read-only systems are a fact of life. But they don't need to be a point of failure.