when i bought my car it ran fine for about 5 minutes then started conking out, turned out to be the fuel pump. For some reason it worked ok when cold but not when hot. I guess that you lose some of the rod stroke when the engine is hot because of expansion of the aluminium block, but i'm surprised it made such a difference. anyway, with a new pump on it was fine.
as has been suggested, you can eliminate the question of a weak pump by putting a jerry can of fuel above the engine and running a pipe from that into the fuel pump. because gravity is now helping you even a duff pump should work ok, that is how i got the car home!