On the first tapping run (on the high E"), I tap with my index finger going outwards...
When he switches to the "B" string, I pull the taps more towards me.
The left hand bridged over the high "E" naturally dampens that string.
This helps me to avoid "sloppying" up the solo while tapping the higher strings...
And, yep, play it slowly at first, then gradually build up speed.
I've also found that practicing certain rock solos on an acoustic helps to make my approach cleaner and aviod the inherent cheating a high-gain amp, low-action guitar can cause.
Good luck!
Lance