Had a similar headache with an old mower of mine. I'd bet it's your fuel pump relay. That's the little switch that tells the pump to run for a couple of seconds when you turn the key. It's probably just worn out from age. Easy way to check is to find the relay (usually in the fuse box), and when you turn the key to the 'ON' position (don't start it), you should feel or hear a little 'click' from it. If it's silent, that's likely your problem. You could also check the fuel pump fuse while you're in there.
Just an idea. Let us know what you find with more photos. Gook luck!