If you want to devote several months of active hunting with nothing else on your agenda, you can buy over the counter licenses for both Oryx and Ibex in New Mexico. Success rates on those licenses are VERY low, especially on the Ibex tag that is for the entire state except for the Florida Mountains.
The once in a lifetime Oryx hunts are not a slam dunk 100% success type hunt that they have been in the past, but still have a pretty good success rate (higher than 50%).
I would say to apply for the Oryx hunt that fits in your schedule over worrying about a specific hunt date over another. Technically the first hunts should have a better opportunity at a trophy animal I guess, but the hunts run for 6 months or so, so if you can't fit the first hunt in on your schedule it isn't really critical.
Rhodes Canyon and the Stallion Range have the most oryx and either would probably be great for a first time non-resident.
The draw odds on the Oyrx once in a lifetime hunts runs around 5% depending on the date and unit and the draw odds for Ibex once in a lifetime runs at about .5%.
Plan on doing a lot of applying before going on the hunt.