Pilot here.

Not easy. At all. The actual flying part? Possibly doable. The basic mechanics of the control system, like the stick and throttle, isn't that different from one plane to the next. You pull back on the stick, houses get smaller. Push forward, houses get bigger. Grab the handle by your left hand and shove it forward and you go faster. Pull it back and you go slower. That's pretty universal to nearly every airplane out there. (For the other pilot nerds, yeah, I know, it's a side by side airplane in the movie and probably had a central throttle. Make you own answer if you don't like mine, I'm simplifying.)

So, flying-wise, maybe possible. The real trick would be starting the thing up. See, WWI aircraft are ludicrously simple. The majority of those engines are piston powered radial engines, which are dead-nuts simple. Introduce fuel, get the propeller spinning, and it'll probably fire up without too much fuss. Shoot, many airplanes from this era didn't even have a throttle. They just ran at full power until the pilot interrupted the ignition system, killing the engine entirely. You controlled your power output by "blipping", which was just turning the engine on and off in succession.

Aircraft from this era were mostly made of wood and fabric, were very slow, and generated huge amounts of drag. They were very underpowered, and the only way you were going to over-speed anything was in a dive or aggressive maneuvering. They weren't stable, don't get me wrong. Not easy to fly by modern standards. Instrumentation was rudimentary at best, and consisted of analog gauges that worked without any pilot action needed. These planes had no electrical systems at all.

On the other hand, a modern jet (1984 is modern enough by aviation standards) is extremely complicated. Just starting the engine requires a very good understanding of what is needed, otherwise you run the risk of a hung start or a hot start, which can cause a fire and completely destroy the entire aircraft. Many of these planes can't even BE started under their own power, they depend on and external power cart for electricity and airflow until the main engine is running. Modern military pilots are expected to have essentially an engineer's level of knowledge about their aircraft systems, because they need to understand what's happening.

Same story with all the instrumentation in the plane. The basic flight instruments would probably work okay, but programming all the navigation equipment requires significant, system-specific training. I doubt that they had enough maps in the side pocket to fly from DC to Egypt by ded reckoning...

Then, of course, there's the fact that even today, not many planes that size actually carry enough fuel to make that trip non-stop. The Smithsonian does maintain some aircraft in a flight-ready status, but they aren't normally stored with full fuel tanks.

Assuming they got everything started, then they still need to be real careful with the flight characteristics. Unlike a WWI fighter, a modern jet is absolutely capable of damaging itself in "normal" flight. Too much power can overspeed or over-temp the engines, causing a failure. Yanking the power back too quickly can cause a compressor stall, resulting in an engine failure, possibly a fire. Pushing the nose down too far can easily overspeed the plane, resulting in structural failure and possibly an in-flight breakup. The stall speed of a modern jet isn't much different than the Cruise speed of a WWI fighter, and swept wing jets have poor slow-speed handling characteristics, resulting in a plane that can easily stall or spin if mismanaged during landing. Many of the critical performance numbers for this type of aircraft aren't going to mean anything to a pilot who has never flown something like that, especially one with only a WWI level of knowledge. (We learned a WHOLE LOT about flying between 1930-1950. Like a truly incredible anount.) Monitoring the mach number to stay below critical, keeping the ITT and EGR where they need to be, ensuring you're using appropriate power settings on N1and N2 for each engine...means nothing to a WWI guy who's never even conceptualized a turbine engine.

Long story short, IF they could get it in the air, a WWI pilot could probably keep the plane right side up, but actually going somewhere is more difficult. The hardest part is getting it up and running without just frying it.