You can catch Feebas with any rod. Personally I prefer to use Old Rod because it makes any "by-catch" easy to kill (usually Magikarp), however many people prefer to use a Super Rod because the only other thing you'll catch with it in that location is a Carvanah and they're also usually quite a high level, so you can train your Pokémon at the same time as searching for Feebas.
Be prepared, unless you're really lucky, looking for Feebas will probably take a long time. It can be found in the river near to the Weather Institute. However it only appears in 6 tiles in the whole river! There is no way to know which 6 tiles, it's random and it's different in every game, so you just have to use trial and error. :( I think the spawn rate for Feebas (on any of the 6 tiles) is about 60%, so you should fish each tile 3-5 times to make sure that Feebas isn't there. I got bored doing that though and just fished each tile 1-2 times and hoped for the best, and it took me about an hour to find him.
If you make a note of where you found Feebas, it will continue to spawn there, until the Trendy Phrase in Dewford changes! (Warning... I think it will change if you mix records.) I recommend that if you want to be able to get more Feebas later, make sure you catch a female, so that if the 6 tiles move, you can breed it if you want another one later. Good luck!