For example http://www.youtube.com/watch?v=BWDvWICcQek that starts at 0:18. When the run goes from the A string to the G string I obviously mute with my palm, but that's not possible when I do it downwards, from the G string to the A string. What I do right now is touch the G string with the ring finger at the same time that I tap the A string with the middle finger and mute it that way.

Is there a more proper procedure in doing this? I can't use the left hand because my fingers are arched.

I apologize if this has been covered before but I could not find a similar topic.
Those string skipping tapping runs he does are incredible. Incredibly hard too. I started working on this about 2 months ago, without having attempted this technique before, and had a great deal of difficulty for the same reasons.
I just jumped on the guitar to see exactly what I'm doing muting-wise on a similar run. On the way up, I'm muting with the side of my palm just like you. On the way down, there is a difference. I'm not arching my index finger so much as angling it. This is a bit hard to describe, but basically I'm fretting w/ my index finger just slightly below the tip, and angling it so that the underside touches the string above it, and just barely touches the string above that.
Other than that it is mostly a matter of fretting and defretting as precisely as possible, and with no excess motion. Work on this one very, very slowly at first, and work it up to speed very gradually. Break the lick down into smaller fragments and practice the frags as well. That was a key for me - having small enough fragments to work on that I could really concentrate on my technique.
All told I think I must have logged a good 25 hrs practice just on the two similar runs in Sea of Lies. They add up to 2 bars (1/2 for the first one, 1 1/2 for the second). 25 hrs on 2 bars. Wow. Hopefully it goes faster for you!
all i can say is it takes ****ing forever to get them clean like MR does.

se012101 hit the nail on the head here, 25hrs on 2 bars. The sea of lies taps arent that hard, but making them completely clean, a different story.
Jackson KV2 USA Snow White/Black Bevels
Yeah I understand what you mean with angling the index finger - that's what I do on the famous Sea of Lies tapping part, but it's harder for me on the Sins and Shadows run. I'll see if I can adjust. And yeah, I've noticed it takes forever to get them clean.

Thanks a lot for the tips!