This actually helps more as it is a real example that uses known game mechanics.
So first, let's remember that you need to pick a leader. Considering that, in this example, you started as Rome and began a cultural game, in the game setup you probably already knew you were going to attempt this. You could unlock a cultural civ in the exploration age just by who you pick. So a good candidate for you is Ibn Battuta. First he automatically unlocks the Abbasids (Cultural/Economic) in Exploration Age. And you will probably like that he will grant you 2 free attribute points after your first civic in every Age. I also suspect that some civics will grant attribute points. So there you have a more flexible leader and he happens to unlock a continued cultural path for you in this example. However, let's ignore this and say you instead chose Isabella. (Because I have a video for reference to use in a worst case scenario example.)
Rome is cultural and militaristic, this gives you 4 custom policies all game long. So to continue to pursue a cultural strategy, you still have access to:
1. Cursus Honorum - Building military units grants

equal to 25% of its production cost.
2. Latinitas -+10%

,

,

in towns with a specialization
3. Princeps Civitatis - +1

on urban districts in your capital
Now, these don't seem like a ton of help individually but they could add up and be useful. You will have cultural attribute points available as well as the ability to have your wonders provide an extra culture and happiness per turn if you spend 2 points on it. Your cultural attribute points will generally let you generate more culture if you invest them there. However, to pursue culture in Exploration Age you will need relics, which means you will want to be playing into the religion aspect of the game. We don't have a lot of information about how this is done but we do know religion plays a role but so do the 'random events'. If you have Isabella to work with, in Spain having an easier time supporting a large navy can help you spread your religion to the new world. (Probably helpful with relics) You do have access to some culture yield output but not sure how helpful that is for relics. I would imagine civic research is a key component though.
So running Latinitas and specializing your towns would help, as would building Casa Consistorial to help push +5 culture in a couple cities. But really pushing for a strategy around relics isn't really feasible until we find out more about that apsect of gameplay. However, your attribute points only help you generate culture. Something you could forfeit in Age 2 if absolutely necessary and build up your economics and science, then swing back into it with Mexico in Age 3. (Guaranteed) By prioritizing archaeology in Age 3 you could beat your opponents there. And if you do you, we established that you got 3 cultural legacy points in Age 1, and you would have to have 3 in Age 3 or else the victory wouldn't even be possible so you would have a minimum of at least 6/9 (65%) legacy points boosting you forward in this example. You could even have 7-9 points depending on what path you prioritized in Age2. If you were the first one to house 15 artifacts in the modern Age by prioritizing, you would most likely take the World Fair Victory even without getting all 9 points.
Now, this all is just staring at limited information of the mechanics and there are certainly things unseen that this example cannot illustrate but you can see how a strategy could be formed within the known limitations that doesn't require you to be cultural. You have always been able to win cultural victories in Civ without having a civ with a cultural trait. Since it is no longer about simply amassing culture points beyond a threshold it has become an even more challenging and flexible victory condition. I am also really glad to see that an economic victory has finally came into the series.