As Stormin suggested I agree you should check the "air gap" on the ignition coil even though it sounds like a fuel problem. I remove the ignition coil every few years and clean the edges and magnet on the flywheel with sandpaper and coat it with a little oil or vaseline or whatever. This helps keep down the rust. I generally use three thicknesses of printer paper to adjust the air gap. Check the spark plug connector and clean out any corrosion and squeeze it a bit with pliers to ensure good contact. Sometimes the spring for the governor will get a bit stretched and can be shortened to where it just barely has tension with the engine off. Try holding the throttle all the way closed while it's running and adjust the idle speed to where it seems about right. If it's too low it may be hard to start. Also, if the throttle/choke cable has slipped it can cause the choke to be on or off when you want it the other way. Removing the head might be in order to--scrape off the carbon deposits and clean the valve seats being careful not do damage the seats. I have had success many times simply by putting some high-temp silicone on the old head gasket and reusing it.
My old B&S lawnmower sounds very similar to yours but it's only 23 years old. My experience with these engines is that as long as the piston is still attached to the crankshaft they can be made to mow lawns. I was having a lot of trouble with mine in the past with hard starting and I even turned the engine sideways so I could pull the rope from the side. This arrangement also eliminated some problems with the cable moving out of adjustment as it eliminated the 90 degree turn in the cable. Also, check to be sure the blade is tight. These things need it to maintain momentum.