The full anecdote as told by Hardy is

> I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” Ramanujan replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.

Hardy just saw it on a taxi cab and expressed sadness; Ramanujam pointed out the mathematical beauty of the number.

The real question is, did he know of this fact already or did he come up with it on the spot after hearing what Hardy said?

I suspect he knew it already. It kind of stands out as an almost-repeated digit pattern if you look at a table of cubes (in base 10), as the cube sums concerned are 10³+9³ = 1000+729 and 12³+1³ = 1728+1. Showing that it's the smallest such number is not difficult, but would take a bit of thought to come up with on the spot. You can do it by checking a few combinations of terms from your table:

   n   n³
  -------
   1    1
   2    8
   3   27
   4   64
   5  125
   6  216
   7  343
   8  512
   9  729
  10 1000
  11 1331
  12 1728