There is a known phenomenon in integer partition theory that almost all integer partitions, after a normalization ($\pi/\sqrt{6n}$ where $n$ is the norm of the partition), have young diagrams which are approximated by the curve $$e^{x}+ e^{y} = 1. $$ This curve is called the limit shape. I was wondering "how large must $n$ be for one to see a close curve fitting?" I ask because I tried observing this by drawing a random parittion of size $n=1000$ and have observed only topologically similarity but the curves did not match precisely.

It sounds like you are asking about large deviations. For partitions, there is a paper by Boris Pittel called "On a Likely Shape of the Random Ferrers Diagram." (1997). This paper contains a proof of the formula in your question, along with bigO estimates about the variability around that curve. I am not aware of any guaranteed error bounds that are not in the BigO format for this curve. Unless guaranteed error bounds exist somewhere for a finite $n$, any attempt to look "by eye" at an empirical limit shape and compare it to the theoretical one is in general inconclusive, since the constants implied by the BigO might be so huge that it crowds out the limiting curve for the particular $n$ chosen. Also, if you are simulating partitions of $n$ uniformly at random there are only a few methods I am aware of that don't introduce bias into the distribution, which I will list here in case the issue is the random generation method.
EDIT: After perusing another post I now recall the large deviations work of Dembo, Zeitouni, and Vershik, "Large Deviations for integer partitions" (1998). This work should provide a complete answer to the question. 

