I worked for a couple of CV580 operations; one was passengers and ran out of Tonga (not good). The other was Kelowna Flightcraft. We also flew B727's (we did one for Greyhound Bus lines as Greyhound Air which was fun) and CV580/5800's, freighters and pax. The one that really got me going was an ACMI we did in the Philippines for FedEx. After that I was hooked on the international stuff. I flew for them for a couple more years in N America, then headed to the Middle East, Central America and eventually the S Pacific, which is how I came to be here.
I had a yes letter from AAI (I have a B747 classic type on an FAA ticket), which has been mentioned, but while waiting around for them to re-organize I came to Air NZ, which was a very good thing. I would have loved to try out AAI, some would have wished they didn't, but again, I have flown out of some pretty third world places, as a base, and loved every minute of it.
Your jet time is good enough for EK by the way (not sure about the PIC but Typhoon pilot would be), do you not want to go to Dubai? Also, I'd get ahold of the people who hire for AAI. Not sure how to start the process these day's, perhaps on their web site, but PM me and I can help out a little better. I can also forward the latest P&C to you.