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In what sense can it be said that spacetime possesses momentum? Can an experiment be envisaged to test this question?

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In the case of an asymptotically flat spacetime (basically, one in which all of the matter is confined to a finite area of the spacetime), there is a quantity called the ADM Momentum that will tell you the total momentum contained in the spacetime, factoring both the matter and a contribution "from the geometry". There are other schemes that let you do stuff like this if you have a timelike or null killing vector defined on a timelike or null 3-d surface. Then, you can talk about the momentum (and energy) contained "within" this surface. There is a review article on the topic here

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