+1 on the Magic Eraser. We don't need no stinkin exorbitantly priced rip off snake oil cleaners 'round here!
Seriously, it works amazing. Use it all the time and stylus is always sparkling clean. It's white so you can easily see the lil blobs of black debris it pulls off. If your hands are steady enough just dab up and down a few times as arm is raised on lifter. Don't lock arm into resting place, just in case you slip. That's my technique, it's quicker and easier than laying it down and dropping stylus on it. Also, if you really trust yourself, you can very very gently shimmy it around a few times on the horizontal plane to get it super clean.
I also sometimes dab stylus a few times with a blob of Blu-tak stuck on the end of a tongue depressor -- or similar. Also works great.
Both these methods are recommended by respected members of the vinyl community: Mark Baker at Origin Live, Peter Lederman at Soundsmith, among others.