Hello from Mexico.Question about the startup

Good evening! I bought a 1982 Shaft a couple of weeks ago and finally got it running after changing the battery, the spark plugs, and adding oil (it didn’t have any). I haven’t changed the filter yet—I just wanted to see if it would run. However, after it finally started up, I noticed that it sounded like it was revving on its own, and I also noticed a very slight burnt smell. It has been sitting for almost a year now, so maybe the smell is due to that. But does anyone have an idea as to why it sounded like it was revving on its own? Could it be a faulty throttle cable? 

Welcome aboard!

This section is for new members to introduce themselves. The rest of your post likely belongs in the carburetor section.

Do you have a genuine Kawasaki factory service manual?

At any rate, the cables could be a problem. A careful lubrication and adjustment should tell you if they are merely sticky, or if they are shot and need replacement.

After sitting that long, the carburetors are going to be clogged up with the residue of old gasoline. If the gasoline where you live has ethanol in it, you can be absolutely certain they are clogged and will need to be completely disassembled and carefully cleaned. They will also need new orings and gaskets.

If the rubber carburetor holders between the carbs and the engine show signs of small surface cracks or if they are well hardened, they will be leaking and can cause the problems you have.

Also, if the timing advancer is gummy or rusty, it may be stuck in the fully advanced position, causing th idle to be too high.