How much of our universe is outside of our light cone today? It seems to me that part of our universe is spacelike separated from us. So it's part of the same space as ours but it's outside of our light cone because of the expansion of space. So there could be no cause and effect between us and the galaxy. So a galaxy from 12 billion years ago today would be outside of our light cone. We would need something faster than light to see it as it is today. The question is, how much of our universe is spacelike separated from us today? Isn't that true for every point in space? So would the Eye of Sauron galaxy which is 62 million light years from Earth, have a different observable universe from that point in space than Earth? Wouldn't part of the observable universe for that galaxy be outside the observable universe for Earth?
 A: 
The question is, how much of our universe is spacelike separated from us today?

There's a cosmic event horizon, defining the asymptotically observable universe. In terms of comoving distance, our past lightcone is already about ¾ there, so in terms of comoving volume, we can already see about (¾)³≈40% of all there is to see (this might only be a tiny fraction of the universe, which could even be infinite in size).
However, this estimate does not account for facts such as that proper volume (at constant cosmological time) increases, that the early universe is opaque to us, that we see distant objects as they were at a certain time in their past (and won't get to see them at all past a certain age), and that observation will become increasingly harder due to redshift and decreasing flux.

Isn't that true for every point in space? So would the Eye of Sauron galaxy which is 62 million light years from Earth, have a different observable universe from that point in space than Earth? Wouldn't part of the observable universe for that galaxy be outside the observable universe for Earth?

Yes.
A: Strictly speaking, nearly everything is outside our lightcone today. Remember that "current" Alpha Centauri is outside our lightcone: only Alpha Centauri 4.3 years back and earlier is a part of it. 
The scare quotes around current also matter - as you probably know, there is no well-defined simultaneity in relativity. However, for matter objects that is not moving at relativistic speeds relative to each other like stars and galaxies this is not too problematic. We can define a cosmological time from Big Bang to now that we would mostly agree on, at least for the purposes of this question. 
A remote galaxy like NGC 4151 sees a slightly different universe than we do. In one direction it sees 62 million light years further than we do, in the opposite direction 62 million light years less (this is a very tiny difference on cosmological scales). However, (ignoring expansion), in 62 million light years we will also see the further objects and NGC 4151 will see what we see. 
There is no evidence that the universe is bounded, so at the very least there are plenty of things outside our current observable universe and our eventually observable universe. 
