don't see how you could get pressure, unless the tank is heating up substantially or your pump was pumping the wrong direction

I think you're misinterpreting the sound .... for starters, the cap should be vented to prevent a vacuum happening, so ya need to get a new cap, but the poor running being the problem, maybe you've picked up some junk from the tank and plugged the carb a bit? got an inline filter? tried cranking with the fuel line off the carb [into a can] to see if the pump has a good volume flow? sure it's a fuel problem, not ignition?
Bill