Yes, I'm in-agreement with maleko89 where the airbox is required to properly-run. That's also been my experience. I can get it to start, but it runs much-better with the airbox and lid secured, especially when trying to see about it running at increased revs.
Replace the airbox and lid, and if you removed the Y-piece of the lid, replace it. Then try to start it.
Have you ever tried using the flat palm of your hand, to 'choke-out' an engine, to see if it runs better? You lay the hand across the carb venturi (the airhorn/top of the carburetor) and then rev the engine. Often the 'hand-choke' will cause it to rev better, a sign that you need to replace the air filter for the induction system to operate as it's jetted. You may also find that one cyl in particular which isn't firing, will 'kick-in,' helping you to diagnose a problem with that particular cyl.