The Arse have got the mighty Villa at home that day; tricky to call demand for that one, I suspect it could be full as they scrap for european places so scalpers could have a premium going on. Keep checking frequently with arsenal website and ticketmaster.co.uk (don't use the .com site, it won't find tickets even when they exist on the .co.uk). Your predicament is made worse by the early K.O - noon. Assuming you wouldn't have any jet lag at all, you'd have to get a flight at something like 9am to be able to make it to Emirates Stadium on time. Getting back isn't an issue, there's lots of choice.
Flights from Belfast are limited, but there's a vast amount of bargains to be had as it's still way in advance and a heavily flown route. I've no idea where the $500 is coming from, someone is trying to take the piss out of you.
Pick from
Easyjet departure 06:35 ex BFS (that's belfast international) to STN (that's Stansted)
flybmi.com departure 06:45 ex BHD (that's belfast city) to LHR (Heathrow)
flybe.com departure 06:50 ex BHD to LGW (Gatwick - not as inconvenient as you'd think for your needs)
aerlingus.com departure 07:20 ex BFS to LHR
easyjet.com departure 07:25 ex BFS to LTN (Luton)
flybmi.com departure 08:45 ex BHD to LHR
That's it. Well it's not, but it is in terms of getting to London in time to transfer across to the game.
There are far too many options to pick for getting back, so you'll have to excuse me if I don't list them - find them yourselves though at http://www.skyscanner.net.
Out of those, Stansted to Emirates is 1hr 30 to 1hr 50 min - even though geographically it's probably the closest after Luton it takes ages - get the train to Liverpool Lime Street (if it's running, check at http://www.stanstedexpress.com), get a bus to Holloway Road Station.
Luton is quicker - around 1hr 30 min - get the airport transfer bus (free) to Luton Airport parkway, get a train to Kentish Town station, get the 393 bus from there to Holloway Road Station and then follow the crowd. Make sure http://www.firstcapitalconnect.co.uk (the train co) don't have any engineering works either.
Gatwick is easy - as no buses are involved which can get confusing - get a train to Victoria (get a normal one not the http://www.gatwickexpress.com as they're only a few min slower but vastly cheaper), get a victoria tube (light blue) one stop to Green Park, get on the picadilly line (dark blue) to Arsenal (which funnily enough is the nearest tube station to Arsenal football stadium). It's about 1hr 10-20 min.
Heathrow would be the ultra boring option - get on the tube at Heathrow - it's the picadilly line (dark blue - no other tube lines serve it) and a full hour later get off at Holloway Road Stadium and follow directions or other fans. You wouldn't see a lot mind of London as most of it'll be underground. It involves no line changes at all though and is slightly faster than Gatwick, so you might wanna consider that. If you're so interested, a tube map is available at http://www.tfl.gov.uk.
Personally, I'd fly into heathrow as there's far less chance of the tube line being shut down for maintenance over the xmas period than train lines (which would involve all the three other options). Pick your airport and find out the transport options at http://www.transportdirect.com. I'd wait until there's 11-12 weeks to go as the stadium tix should be on sale then, plus the engineering timetable for xmas should be out so you can figure out which trains are or aren't running, yet the flights will still be easily affordable.
