How can an observer observe the metric of spacetime? I don't mean how can we measure the metric in practice. I only mean in principle. Suppose you are an omnipresent being, no experimental limitations. What measurements do you need to measure the metric at a point?
Also, you can deduce the metric from the Energy Momentum Tensor, sure. I don't want that answer. The answer I want is "How can we deduce the metric by making measurements about the geometry of spacetime"?
 A: Directly measuring the geometry is not practical, as the measurements would be too hard to make. But it could be done in principle. You just need to parallel transport a vector around a loop and measure how much this vector has changed. The change in the direction of the vector is directly related to the Riemann tensor.
Suppose you take a vector and you parallel transport it round a small square:

Obviously when you finish going round the square you'll be back at your starting point and the vector will still be pointing in the same direction.
But this is only true when the spacetime you're moving through is flat. If you try this on a curved spacetime you'll find that after completing the square you won't be back at your starting point and the vector may have rotated away from it's original direction.

In this diagram the red vector is the initial vector at $P$, and we find that when we get back to our starting point, the angle of the vector has changed.
The change in the vector is described by the Riemann tensor, and in General Relativity the Riemann tensor is what determines the metric. So we can measure the metric by measuring the geometry.
In practice, the changes around any loop of a practical size would be too small to measure so this direct measurement could not be done. Instead you would do indirect measurements. For example just dropping an object and measuring its trajectory is a way of indirectly determining the metric since the acceleration is determined by the Christoffel symbols, which are determined by the metric.
