I've been playing for years, and I can't even nail the song with the proper tempo and everything. Don't worry, whether you suck or not, I suck even worse...
Yeah, for 'Master of puppets', it's all down-picking. Down-picking very fast and alt-picking sound different in my opinion... I've tried alt-picking, but I find that it's difficult to have consistency when alt-picking that slow. 'Master of puppets' seems to be right in the middle. Too fast for down-picking, and too slow for alt-picking.
My advice would be to keep working at it. Don't take shortcuts, use down-strokes all the way like on the record, but start slow and slowly build up. Don't play along with the record just yet. It takes a lot of time and practice to get that fast. As for me, it doesn't matter since I don't play metal anymore. But for you, keep at it...