You don't. That server is a security nightmare.
Judging from the kernel, you're running RHEL 6, which was released in 2011, 13 years ago.
- Full Support ended in 2016.
- Maintenance Support ended in 2020.
- Extended Life Cycle support ($800/year, doesn't cover everything) ended this year.
Your exact kernel version, 2.6.32-431.29.2, was shipped with RHEL 6.5, released in 2013. This suggests you're missing about 10 years of security updates.
On top of all this, RHEL 6 did not ship with Apache 2.4.37 (which was released in 2018), so this is a custom install. Depending on how that was done, it's possible your Apache has never received security updates.
Then there's the sites running on your server. If those are dynamic sites, perhaps using frameworks and databases, then those are also ancient, out of support, and at risk of attacks.
Your server is running ancient software that is 100% unsupported. It has not had security updates for about a decade, will never receive any new security updates. It is done. Upgrade to a more recent OS.
Let me be very clear:
It is irresponsible to keep this server connected to a network. You need to rebuild it on a modern, supported OS.
It also has a lot of valuable content with a special setup that I don't want people to mess around with.
... but upgrading means doing backups of gigs of data and basically wiping out the whole OS, something that may be time consuming and costly while potentially putting on downtime for the website guests. There has to be a way to update openssl without redoing the OS.
I feel for you, but this is the reality of (hosting) software. It needs maintenance. You cannot just leave it alone for 10+ years and expect it to be fine.
Your current situation is unsupported and insecure. This is not just about OpenSSL. The server will also break more and be increasingly harder to fix. Your comment about having to make backups also worries me. If this is important, those backups should exist already.
Rebuilding is just vital maintenance at this point. It should have been budgeted for. If not, it must be budgeted for now.