[Yoyizit]

I pump the primer 5 times, and it starts every time, but just as the engine is reaching normal rpms, the engine dies. It only runs for 1-2 seconds then dies.

It sounds like you are running on the gas in the float bowl, if it has a float bowl.
There is a fuel line obstruction upstream of the primer bulb?

 

[Yoyizit]

Electrical problem/ignition/ignition timing?

I don't know if these engines have a timing advance, either vacuum or centrifugal or both.

 

[Yoyizit]

I don't know hardly anything about electrical nor ignition systems. If I knew what to look for, I might be able to figure it out, but I don't even know where to start to look for problems there. Any suggestions?

You'll need a schematic. There might be a generic one online.
I'm thinking now that the engine vibrations open up a loose connection somewhere in the ignition circuitry. The engine stops, the connection closes again, and the cycle repeats.

If it is a fuel problem, what quantity of fuel is burned in 2 seconds? If it's not the gas in the float bowl, what other volume of fuel is being used up this quickly?
If anyone can post what running time they get from a full fuel tank and the tank volume, we can figure out what fraction of a cubic inch you are looking for. Maybe an air bubble forms somewhere because some air bleed opening somewhere is clogged.

 

[Yoyizit]

there is more fuel in the float bowl that could burn in the short time this thing runs.

The governor should even out the speed with varying demands on the engine, rather than kill it.

Post what technical info you have on this machine and we can do some informed troubleshooting.

Loosen the gas cap and see if it still stalls.

 

[Yoyizit]

Forgot; check for particles in the gas tank, flush if necessary.

Yes, whatever tech. info on this that you can beg, borrow or steal, online or off. Sometimes I look in library books on small engine repair for ideas. Occasionally a book will mention some detail that the others omitted.

 

[Yoyizit]

You can check that you have ignition by putting a spark gap in series with your spark plug.

If you use another spark plug for this gap, close it down to 0.01" or so, about the thickness of 2 sheets of copy paper.
To withstand engine vibrations you will have to mechanically secure the extra plug to the engine housing with insulating material that can stand a few KV. Dry wood comes to mind.
As long as you see a spark in the tiny gap you are getting ignition.

Alternately use clear tubing, epoxy and parts from other spark plugs to make your visible spark gap. This will be smaller and lighter so it may stand on its own. You could use a 6-32 machine screw sharpened to a point to mate with the screw-on connector on top of the plug, or use paper clip wire [which is springy]. The current through this gadget is only a few mA.

An old neon timing light may also work, but the ignition voltage for a small engine may to be low for this.

I've never had to make one of these gaps, not yet. . .