Quick Edit for ratsmilk's post that wasn't there when I started typing:
Arena Trap will not work on Mesprit because of Levitate :X
Tips for catching:
#1: Don't use fly, ever since roaming pokemon were introduced in Generation II, they have randomized their location every time you use fly or any other form of transportation besides running [or biking]. They also randomize to any route you are not currently on when you load a save file so you can't simply get to the route it's on, save and keep re-trying.
#2: Since gen II, the roaming pokemon will always change routes every time you do, but will almost always move to a route adjacent to the one it is currently on. They do occasionally re-randomize, but not very often at all..
#3: You don't need mean look, you only need a pokemon faster than Mesprit, a Wobbufet and some patience.
Strategy for hunting down Mesprit is to abuse the fact it teleports every time you use Fly by flying repeatedly to a town with as many exits as possible. Doing this will increase the chance of Mesprit randomizing to a location near you. After flying many times and finally getting Mesprit near you, check carefully which route it is on.
- If it is on a route directly adjacent to your town, exit through any of the routes Mesprit is NOT currently on and check the map again to see if it moved to the route you're currently on.
- If Mesprit is on a route nearby but there is one route in between you and Mesprit, move to the route between you and it. There's a 50/50 shot it move to that route when you do.
Strategy for actually catching Mesprit once you're on the same route is tedious without Mean Look, but not nearly as bad as other roaming legendaries with Roar *coughENTEIRAIKOUcough*. You'll want a pokemon faster than Mesprit so you can get an attack in before it runs, preferably False Swipe if you have anything that can learn it. If not, Mesprit's defense isn't too terrible and it shouldn't be too hard to damage it without killing it.
If you don't have any pokemon faster than Mesprit, then you have three options before you:
- Use a Quick Claw as a held item for a 10% chance of striking first regardless of either pokemon's speed stat. I'd suggest using a paralysis-inflicting move if you go with this method since Mesprit will have its speed cut drastically once it is paralysed. If you don't have a Quick Claw, you can pick one up from wild Sneasel near snowpoint
- Use priority attacks such as Quick Attack and Aqua Jet which will always strike first. You won't do much damage, but then there also won't be much risk of accidentally KOing mesprit until its already on low health.
- Use a Wobbufet [at least level 30]. Wobbufet will trap mesprit with its ability Shadow Tag and can [slowly] damage mesprit with Mirror Coat. I would not recommend this strategy because Mesprit only has one move that Mirror Coat can reflect and you run the risk of depleting its PP and causing it to struggle to death.
Once you have Mesprit down on health [and preferrably either asleep or paralyzed], save every time you enter the same route as it before trying to engage it. Move a wobbufet to the front of your party, walk through grass until you find mesprit and then hurl balls at it until you run out. If you run out of balls, mesprit escapes or anything of that sort happens to end the battle, turn off the power and restart to get your balls back... just remember that Mesprit will have randomized its location again.
If you don't have a Wobbufet then Quick Balls are your best hope at this point. You'll just have to track it down repeatedly and hurl one per battle until you catch it.
[Apoligies for any inconsistencies with the capitalization of "Mesprit"]