A library is enough to hold a first ring and prevent city flip even if there are a number of nearby high culture opposing cities?
If the library isn't enough, the monument isn't likely to change things.
If the tile you are fighting over is on the outer ring of your culture
and the outer ring of the AI's culture, you should probably just assume you're going to lose it against a high-level AI. AIs love building culture buildings (especially when you start trying to amp up cultural pressure against them yourself), and get a hefty discount on the cost of making buildings at higher levels. And they'll even bump up the culture slider or run artist specialists sometimes. Assume an AI city will produce at least a little more culture than your city, unless you are specifically trying to get that city to legendary culture for cultural victory. Trying to beat them at their own game by just "build more culture buildings than them" is a losing proposition - sure, you might be able to
do it if you prioritized it heavily enough, but you'd have to dump a ton of hammers into something not all that useful. It would certainly take a lot more than just a library and a monument. You'd need to build theaters, temples, monasteries, wonders.
But that doesn't mean the AI cities will simply run rampant over your culture. If a tile is
closer to your city than it is to the AI city, then as long as you can stay on the same cultural
tier as their city - both Fledgling 10-99 culture, both Developing 100-500, etc. - your tile culture is getting a hidden under-the-hood 20 culture per turn advantage. It's not your city with 2 culture from a library up against their city. It's your city with 22 culture, 2 from the library and 20 from the interior culture advantage, up against theirs. Adding the monument for 23 culture per turn instead of 22 isn't the big deal. Even if you only spend half the time on culture-tier parity, that's still half the time you're getting an extra 20 per turn; a city with a library can probably fend off an AI city making 8 or 10 culture per turn. Adding a monument occasionally will shift that balance for a specific tile, but not often.
So basically, if an AI city isn't producing boatloads of culture you can probably defend tiles you care about with just a library. And if it
is producing boatloads of culture, rather than fight head-on against AIs at something they care far too much about and get far too many built-in advantages trying for why not just get an army and take the city away from them? Problem solved.