For light dirt-magic eraser original and a good brisk buff with soft leather or a brown paper bag.
For heavy dirt- magic eraser original with a little alcohol (let it dry) and a brisk buff with the above.
There are a number of products made for shaft cleaning but I have never used them. This works good and it's cheap.
Magic Eraser and alcohol is what most, including repairmen use.
Sometimes, when I get a shaft that is particularly dirty, I will soak the eraser in ammonia. I keep a soft toothbrush to do a little deep scrubbing too. A soft bristle brush won't hurt ya.
I get very dirty shafts most times and you can even use Acetone with the eraser. It will absolutely clean the dirtiest of shafts whistle clean.
I also use Acetone to unstick my sometimes clumsy fingers that I glue together.
Just be careful not to get it close to joint collars or ferrules as Acetone can and will melt just about anything. Some cue materials are safe but why take a chance. I just stop short of a collar or ferrule. Or you can tape them off.
Also, do not use Acetone on any laminated shaft as it will melt the glue on the laminations. Anything else is safe.
But the old Magic Eraser and some elbow grease are the best cleaners.
Paul Fanelli a cue maker recently passed away used Mothers Chrome Polish on a soft rag....Cleaned like new with smooth finish like glass.........Ive been using it for
years
If your shaft is aged and well broken in you can use a couple of squirts of 409 on a paper towel and that will clean off the gunk without removing the patina.
Best thing to do is find a good repairman and have him clean, seal, and wax your shaft,,,,,,,,,,it'll stay clean with just a dry rub down after each use.