If you are spending all this time and resources worrying about the culture from your vassal, you are not focused on the important things. Your game has essentially become only about preventing a few cities from revolting.
Generally, if I take a vassal, I want him to be somewhat productive either militarily or research wise. I at least want him to have 3 or 4 cities.
The question you ask yourself is whether all those cities would be better in your hands, or whether having a good vassal is important. There may be wonders that you capture that you want to keep or just good land.
If you capture some cities and vassal the guy, you can usually tell immediately if a city or two will be under cultural distress. If your Empire is otherwise nice, large and productive, just gift those cities back to the vassal. He will be better for it.
The later the game, ofc, the more issues with culture arise. Cities you capture that immediately start with 0% of your culture are likely to be under distress if still receiving culture pressure from nearby vassal cities.
If you are intent on keeping these cities for yourself, the following notes may help:
1) Don't let these cities grow for the time being. In fact, it may be best to let them shrink (sometimes they are starving initially anyway), or whip a few culture buildings in to shrink it and get some culture flowing.
3) Obviously, the more units in the city reduce revolt risk. But also the type of unit helps. That is, how advance the units are. So upgrade units or put better units in the city helps. Ofc, this costs your money and manpower that could be spent killing more enemies elsewhere.
4) Get some culture buildings in (theatre, library, monastery), run artist, build culture. Culture grows very slowly in this game, but once you hit the 50% mark there should be no more revolt risk.
5) Probably not worth it to build hermitage, cathedrals in these cities unless they start going into negative culture growth
6) Again, ask yourself if this is all worth it. You've left Capac with one city - his capital - which has huge culture pressure from being the cap and wonders/other buildings. Give him back some surrounding cities and those cities will have far less culture pressure. He will contribute more to your game.
Another way to look at this, is that you are spending all these resources preventing revolts in city that are basically doing nothing for you. If you are just going to vassal an AI and leave him once city that is going to keep the cities you kept in such a state, you are far better off just taking him out completely..and getting those wonders for yourself.
edit: I actually looked at your game and have a few observations/notes:
1) Vassaling Capac in 1070AD is far from "early". That's considered quite late actually. He already had 6 high culture wonders in Cuzco with another one soon to finish. Wonders that would be much better in your hands. By 1070AD his cap will have a lot of built up internal culture. And Tiwanaku being a "must keep" city. I would have killed Capac. You did leave him with 3 cities, but he ain't doing much for you. You are wasting to much resources worrying about this culture mess
2) Keep tabs on resource trades. You are missing 34 gold per turn right now. Don't let that happen
3) I see no goals here with your game. You've played this game very late now, but don't appear to be doing anything. If going for Space, you haven't even built Oxford yet in your capital, which should have been done at least a 1000 years ago. Maybe you are just tooling around for fun or very new to the game, but I see no purpose here. Meanwhile certain AIs are just getting stronger and stronger.
4) That HRE city is hilarious. Sury plopped down a few cats on that ice island and BK's LBs are at minimum strength, but he can't take the city without some attack unit ..ha
