It sounds like the air is not getting into the pump on the upstroke. Try unscrewing the collar at the top of the pump tube, pulling out the whole rod with the plunger (leather washer or diaphram) at the end. Give the plunger a massage and a little oil. When putting it back into the tube you must carefully fold the circular edge of the plunger forward all around and do not let it fold back as it goes in. This applies to typical pumps.
Some pumps use little ball bearing valves that sometimes stick, specially if they get rusty. If you can see the ball valve, prod it loose and give it a drop of oil. Otherwise just put a few drops of oil into any holes where you think the ball valve might be and see if the pump begins to work after some exercise.