If an airliner disappeared in 1966 and landed in London 50 years later, without anybody on board being aware that they had been 'lost', what difficulties would they face? I'm thinking that when the aircraft took off, JFK was president of the USA and Obama had been around for 8 years. £1 would buy £2.79 dollars. Avg 3 bed house in England was £3500 on take-off but £200,000 on landing. There were no mobile phones and smoking was widespread, even on the aircraft, but when they landed they'd see British police with machine guns and whatever the event or people they were planning to see, would be history. I omitted to say that they had no idea that they'd been missing for 50 years and hadn't aged. Everyone on board would have been deemed dead, long ago. Hope this helps get the creative juices going.
Instant celebrity and demand
A verifiable magic event has taken place for the first time in Earth's history, and these people are the main cast of this drama. Everyone(!) is going to want a piece of them. Media to begin with, but also scientists of all sorts. Anthropologists, psychologists, and the medical scientists will have a field day. And maybe even the military.
This will probably be very taxing on them. Not only is it hard to be in huge demand but the strange formsof demand will be totally bizarre to them.
"You want me to do what? Pose for that filthy rag?! How dare you!" * SLAP *
The deaths of loved ones, changes
Fifty years is a long time. The grand-parents and probably a good portion of the collected parents will have passed away. Children will have grown up. Some cities will have withered away and hardly even exist. And those that still do exist will have changed.
As stated in other answer/comments: the culture shock. The space race and the cold war is over. Gender roles have been completely upended. Religion has been severely downplayed.
Decency standards have become relaxed in some areas and become incredibly uptight in others; especially the attitudes towards minorities have been severely altered. For instance: to them using any of the N-words against African Americans is still kind of OK, especially "Negro", but the rest of society will gasp in horror as soon as they do.
I think you can expect a great part of them try to find solace in church. And you might expect people to seek out the more strict and conservative churches, as they have attitudes that are more in line with their own fifty years outdated dittos.
A very unique problem, but these are people that have passed out of all the systems and need to be introduced to them again, as adults. There will be some very tricky bureaucracy to deal with as they will — essentially — have to go through the immigration process of their respective countries, while being citizens to begin with and having identities, but that are marked as "deceased".
Providing that DNA testing proves that they are indeed who they say are, there will be some pretty taxing issues about property. Possessions will have been inherited, sold or thrown away. Real-estate will have new tenants. There are no legal precedents here and so they will most likely be left without anything they owned, getting back only that which a) has not become obsolete and junked b) the new owners are generous enough to give them c) they were carrying with them on the plane and that has not been ripped apart and atomized by all the scientists and souvenir hunters.
Or there could be some fairly amusing court battles.
Providing for themselves
People that fly on a trans-atlantic airliner in 1966 will be upper middle to high class. The question is what skills they bring to provide for themselves. When the money from the initial fame runs out.... then what?
This will be a lottery. Some have skills that are time-less. Other have a skill set that has become hopelessly obsolete. Those that can learn fast and adapt will re-learn. Those that do not have that ability, they will have a bad time. Upper class house-wives... oh dear, they will be in for a shock.
And do note they all have a long way to go. They have — for instance — never seen a personal computer or a mobile phone. Interactions with information systems that we all find natural...
— No, you cannot pay with coins on the bus... you need to get a travel pass.
— Ok, where do I get that?
— Over by that booth
— Eh... ok, how do I use this?
— You press the screen to start
— Like so...
— Oh, you mean "poke". Ok... now what. And where do I put the coins?
— You do not, you need to use a credit or cash card.
— Where do I get one of those? Do I go to the bank?
— No, you just order it on-line...
— "On line"? On what line? A bus line?
...will be a complete mystery to them.
Culture shock - things are a lot busier/faster than they used to be. They'll get over it.
Their own deaths - they'll have been pronounced as dead at some point soon after the plane disappeared in 1966. There's going to be some paperwork to go through. Since everyone on the flight will have their passports, at least they'll have some form of identification
I know a few people who still act as though it's the 60's.
I suspect that as soon as anyone realized what "really" happened, the military would be all over this and media or anyone else wouldn't get within a mile of these people or the plane until the military determined whether or not there was any possibility of making a weapon out of this somehow. (Or don't you watch today's paranoid TV/movie plots?)
Merging existing answers, the priority for the world is going to be finding out what happened, in a way which is safe for the world.
None of those passengers are going to get out of a top-level military biohazard containment facility any time soon. The only two possible causes are an unknown scientific phenomena (magic indistinguishable from, etc.) or alien (non-human) intervention. It's not too strong to say that the future of the world depends on making sure we're safe from any threat they may pose.
Followup flights and research along the track of the plane, detailed examination of the plane down to molecules, charge and magnetism, all that kind of stuff is a given.
But the obvious question is are they who they say they are? A new scientific miracle would be amazing, but an alien invasion would be equally plausible. There's no context for something like this. So those people are going to be going through batteries of tests, for at least months and probably years, to find out exactly what they are and what they know. And if any of them happen to die before they get out (and I wouldn't bank against that being non-accidental) then the autopsy is going to take that body apart to component cells.
Assuming they're actual people, by the time they've been imprisoned for a couple of years or so, they'll have had plenty of time to acclimatise to the new world they've arrived in.
Not an airliner, but the Torchwood episode "Out of Time" in series 1 has a small aircraft with a few of people on board get lost in time and arrive in the present day. From memory the craft might of been from the 1960's or 50's.
Characters had to deal with trying to integrate into a society that was completely alien to them. Their families and friends had died and they had no skills to get real jobs etc.
This has already happened†.
Even when the responses to the drug were positive, patients were not always able to cope with the consequences. Rose R was struck by sleeping sickness at the age of 21 and awoke in 1969 to find her world of 1926 had vanished. She remained rooted in the 1920s and, as if the time gap was beyond her comprehension, stopped responding to levodopa.
† Kinda. A large number of people were infected by a disease and were rendered catatonic for decades, after which they were awakened by a drug.