Nice idea!! But you have it the wrong way around. The hair grew on your arm because under the cast there was less rubbing of the skin and so the normal friction induced hair turnover didn't occur. People who have there leg and foot in plaster will also notice that the skin on the sole of their foot gets much thicker because of the same process.
So to make your idea correct - you actually need to put a plaster cast on your top lip! This is unlikely to stay put so you actually need to encase your whole head in plaster - for about 6-8 weeks!!