Metro-Rentals is a good place to start. I also like Hotpads.
Victorian Village, the Short North, and Harrison West will be relatively close to South Campus but sufficiently far to avoid a substantial undergrad presence (and the undergrads who are there will be the ones who moved down that way to get away from all the other undergrads, the campus bars, etc.).
The school is at the corner of High & 12th (or 11th--takes up a block). If you're looking to be able to make it an easy bus ride, look for places along the 2, the 18, and the 84. The 7 will also drop you off in the general neighborhood, though you'll still have another 4-5 blocks to walk to the school.
The 2 will take you down into the Short North, Victorian Village, and Italian Village (as well as Downtown proper, though there are fewer rentals there). It will also take you up towards Clintonville (which includes Olentangy Village). There are fewer apartment complexes up that way, however.
The 84 will take you into Grandview Heights (and the "Fifth by Northwest" area of Columbus that often gets lumped in with GH ... a whole different topic), as well as into Upper Arlington. Both are nice areas. Be warned, though: the ride on the 84 from from there to Drinko is long.
The 18 will take you up Olentangy River Rd., where there are some apartment complexes that are probably a little more economical (a little more suburban, to be honest), and also down Neil Ave. (which IMO is the prettiest street in Columbus when you get south of 5th).
I'd personally look at the Short North, Victorian Village, and Italian Village, simply because the #2 is so direct. If you're in Victorian Village, you also will likely be within walking distance of both the #2 and #18, which gives you some options.
For the specific apartment complexes you named, Arena Crossing is in the middle of the Arena District, so the noise level may be a little on the high side if you're going to by studying at home a lot. (If you're going to do your studying at the school and want to go out at night, though, you could have a lot of fun living in that area.) The Gateway is not going to be good for you if you're trying to stay away from undergrads. They might not live in the complex itself, but they'll be swarming all around it. (It does have the advantage of being a 5-minute walk to class, though. Trust me. That's an asset that you'll appreciate midway through 1L first semester.) I don't know a great deal about the Meridian, but I do know that it would involve a bus transfer to use COTA to get to class from there. The #5 along 5th to High, then the #2 up High to the law school. Short rides on each, true, but the medial wait could be an irritation.
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