Some brands of TV's are better than others some will have a better picture quality etc. Some have the auto power feature, some don't some stay on the s-video (aux 1) channel when you kill the power, some don't. But here is my suggestions, take them with a grain of salt as I'm sure others have other opinions.
1) Buy any brand of TV and go cheap, but get a 27" TV. Vertical games display on a 27" TV are almost the exact same size as they are in the arcade.
2) The only reason to get a more expensive, higher resolution TV would be so that you can navigate windows better. I got a cheap refurbished 27" RCA with svideo for $249.99 (Canadian$$) and arcade games look absolutely fine in my mind. I even compared with some games in the wild and I can't tell the difference. Note this TV doesn't come back on after the power is unplugged for more than 10 min BUT it does stay on the svideo channel so I can work around the power issue, but changing the channel back to svideo each time would have been a pain in the neck.
3) Test that the power will be restored to the TV after unplugging and pluggin it back in it in the store before you buy. Also make sure it stays on the AUX1,2 or svideo channel as well.
4) Get a good video card I recommend any of the ATI radeon (get a built by ATI card) and a good thick short svideo cable and you will be happy. Navigating windows is doable on my setup, but really only at 640X480 and in a pinch at 800X600. Otherwise the games are great! I couldnt' be happier.
No worries, $350 is a respectable amount to find a decent TV just shop around. I got mine at XS-Cargo (refurbished electronics wholesale place).