Is there a traditional card game with similar rules to "Lost Cities"?
I don't think I've ever heard of such card game. To me it's pretty clear that Lost Cities were developed and designed using standard card deck, but I don't suspect it's actually based on some truly similar traditional card game. Maybe the basic idea was to design a 2-player game with simple rules, something a little like Solitaire, but competetive, with  piles shared between the players? 

I heard Lost Cities being described as 2 player Solitaire with an extra portion of suspense - source. If there was more similar game to Lost Cities, maybe such descriptions would mention it.

I guess this isn't the answer you were looking for, although I can't give you any game title that would have the feel of Lost Cities. This is the interview with Reiner Knizia in which he mentions origins of Lost Cities. You also may want to take a look at this interview and this one.


  It started out as a two-player card game. I then tried to develop it into a larger game that would accommodate between two and four players. In the end, I came to the conclusion that I wanted to leave it as I originally had it [as a two player game]. When I start out with a game, it is usually unclear where I will end up with it. If you build up internal obstacles and say "this has to be a card game" or "this has to be a tile-laying game" then you restrict yourself. I think you need to do the game justice -- you need to let it develop. Lost Cities developed into a two-player game: it is clean, it is pure, it is simple, it is challenging, and it is fun.


You might also try checking Patience category on pagat.com