I think I've described how they work:
Establish them as soon as you can afford and close them as soon as fuel increases to high. If fuel decreases enough, you could establish them again.
The 777-200LR really sucks. That's why I prefer doing ULH with normal aircraft, maybe with a tech-stop. This gives me the possiblity to utilize them otherwise if ULH stops to be making enough profit.
If you want to use 777-200LR (or A340-500 etc.), you need to purchase them. This at least minimizes the leasing costs when you are not able to find them.
But, again. The problem isn't only limited to the ULH. ULH and very short small jet routes are the first who suffer from high fuel prices, but if the fuel price is high enough, even your normal long-haul flights suffer.
Key feature of AirwaySim is to know when a certain aircraft stops to be a good one - for example DC-10 is very popular in ATB beginnings, but it stops from being this popular soon, while it lasts most DotM. Same for 727(-200Adv) in DotM. You can make endless money if you choose it first, but when you fail to get rid of it right in time, it will cost you money.
ULH can make profit. You simply need to know when to stop and it's easier if you use multi role aircraft like 777-200ER, 747-400(ER), A330-300 or DC-10-30ER, maybe with tech-stop. As you pointed out there's nearly no competition and the lower ticket price for tech-stops isn't this hard. On the other hand you could use an 777-200LR also on shorter routes. It consumes less fuel than the 777-200ER and is just more expensive to order.
Oh, and I prefer to do the opposite of what Swiftus says. In fact I would prefer the help from my eight year old niece who never has heard of AWS... both will give better results.