I don't mean to sound flippant, but if I walked into the situation you describe, I would immediately walk right back out. That is a a situation set up from the start for failure, and no way would I care to be responsible for it.

I certainly get that we inherit less-than-ideal systems from time to time; I've been there. But I've also learned that every time I get paged in the middle of the night, it's my failure, whether for a lack of an early-warning system, or for doing a bad up-front job of building self-healing into my systems. If I inherit a system that I can tell is going to wake me up at night, I refuse to be responsible for it in an on-call capacity until I've mitigated those problems.

There seems to be this weird thing in the dev/ops world where it's somehow courageous to be woken up at 3am to heroically fix the system and save the company. I've been that guy, and I'm sick of it. It's not heroic: it's a sign of a lack of professionalism leading up to that point. Make your systems more reliable, and make them continue to chug along in the face of failure, without human interaction. If you have management that doesn't support that approach, make them support it, or walk out. Developers and operations people are in high enough demand right now in most markets that there will be another company that would love to have you, hopefully with more respect for your off-duty time.