I don't think you can actually catch Mew. You only need to go through each bonus stage twice and then be at Indigo for a chance of it appearing; but hitting it with your Pokéball doesn't cause the CATCH! to light up, and the best you can do is jsut keep hitting it until time runs out, at which point it will be added to your Pokédex.
Ah, well I have 3 sources, one says 3 times and two say only twice, so I'm guessing the only twice is correct (which is a relief because just beating Mewtwo once is ridiculously hard). As for whether you can actually catch it or not... I've only got two sources that say whether you can or not. One says you can catch it if you hit it like 20 times, the other says it'll never be caught, but will be added to your Pokedex anyway. I think you MIGHT actually be able to catch it, but thankfully it's not required for a full Pokedex entry. If it was and I got to Mew but didn't catch it I might just throw my Gameboy Pocket into the wall.
I've played the Ruby & Saphire one. It was LOADS of fun.
Countless hours of my childhood.
I also have played the RS version a lot (probably more than the original one). I think I have better scores on it and maybe all the Pokemon... I don't know, I'd have to check. I haven't played it in years in any case, not since I got really good at it and could keep going for hours. It's definitely the easier of the two games, because a lot fewer of the elements are random and the tables are better designed (leading to fewer unstoppable losses when the ball happens to bounce just the wrong way). If we were going to do a score contest I'd be for this one over the original, since there's more skill and less luck. And also most of the points for bonus stages are only given AFTER you complete them, so you can't just just barely not beat the highest bonus stage 30 times and get ridiculous scores like you can on the original (Mewtwo stage gives like 500mil-1bil every time you even TRY it, so if you're careful to never beat it you can rack up so many points).
I think after I finish the Pokedex on the old one I'll probably find the RS one and complete it too if I haven't already. It's technically my sisters' game but she never plays it soooo I should be able to do whatever I want with it :P