Although many bars over here would show football if one asks, it's usually only when matches are being shown on one of Fox Soccer networks or if ESPN 2 is showing Prem games. I'm pretty sure FA Cup rights in the states belong to Fox Sports and if Fox Soccer Channel/FSC Plus/Fox Deportes is showing the Notts County game, then most likely you should have no problem watching it in one of the places on the list that okstate99 provided.
However, as of now, the Notts County game is being listed as being shown only on the FoxSoccer.tv website and on Setanta Canada. If that is the case, then your best chance (outside a cybercafe) would be Churchill's Pub in Miami city proper, they alone would be the only place in Miami that might carry the Canadian Setanta feed. I've also heard of times where they projected streaming online video to a large screen, but that might be unlikely for Notts County.
Additionally, because of the proliferation of outlets to watch English football, Churchill's is no longer the "go to" place that it used to be -- case in point, I went to see the derby there one morning with a friend of mine and we were the only ones there, aside from some guy who was from Stoke-on-Trent and just happened to be there. Years ago, there would have been a lot more people there (rags mainly, of course) and now the emphasis on showing matches has been relegated as the pub focuses on being a venue for live shows at night. The other problem is that Churchill's is located in a dodgy neighborhood (Little Haiti) but I would say that it is not as bad as it may appear and during the day its no more dangerous than any of the more "touristy" neighborhoods.
Because of this, I would check with this website to see if Fox decides to broadcast the Notts game on one of its soccer networks first:
http://www.livesoccertv.com/competitions/england/fa-cup/
And if they are not then I would check with Churchill's to see if they are showing the game. I wouldn't check on Churchill's sports schedule on the website however -- they rarely update it!
Phone number is on their website:
http://www.churchillspub.com
Otherwise, online might be the only option unfortunately.
Anyway, the Villa match has started, I hope this helps!