Great question and some great answers so far.
The flying part was not necessarily that hard in terms of technical airplane-steering skill versus the norms of pilots of the day.
Airplanes of the day were underpowered and not very stable. And for a long range flight, tended to be grossly overloaded with as much fuel as could be crammed into nooks and crannies. With the result that the engines and wings were just barely able to keep the thing in the air for the first couple / few hours. Better fly precisely; every wander you permit increases drag. Do too much of that and soon you’re skimming the wavetops.
Creature comforts were non-existent, and flights long, so personal mental and physical endurance was huge. Like a climber getting halfway up a tall cliff, now you’re going to have to push yourself hard to succeed or die trying. And both outcomes were common in early aviation.
Lindbergh briefly fell asleep a couple times on his journey and awoke out of control. But not so out of control that he could not recover before impacting the ocean. One more, or slightly longer, sleep episode and he might have been another case of “intrepid flier vanishes into the sea”.
As suggested upthread, the truly hard job was navigating. Not easy over mostly uninhabited land where there were no good maps, and even harder over water.
The precision of your own compass(es) and airspeed indicators was not great. At least by then good chronometers were available. Your ability to assess wind direction and speed was near zero.
If you, like Earhart, have a dedicated navigator on board who has time to use a sextant, and the sky above you is clear enough, you can get fairly accurate position fixes, albeit after a few minutes of math. But with less than great maps, knowing that 15 minutes ago you were at 23.456N 160.678W may not be as informative as you wish it was.
Successive fixes can give your actual direction and speed of progress over the surface. Comparing that to the compass course and speed you’ve held gives insight into the wind. But the error bars on all this stuff are large. Two fixes 30 minutes apart should show you having covered 45 to 60 miles at the speed of early airplanes. Slight errors in sextant work means each position is really a circle a couple / few miles in diameter.
You can draw a lot of different straight lines between points picked inside two e.g. 5" diameter circles 50" apart. One of those lines is real, and all of the other dozens of possibilities are mistakes. What you deduce about the winds depends critically on picking a line close to the real one. Plus you need to have flown consistently as to airspeed and course the whole time. If you’re wandering around on either parameter, that just increases the error bars on your wind calculation.
So finally your navigator maths up a wind number you can believe. Which was the average wind you experienced from ~15 to ~45 minutes ago. How predictive is that of the next half hour?
Weather science was still in its infancy. Sure, ships had been plying the seas for centuries, but their low speed meant all weather was local. Even at 100 knots, a plane was 10x faster than a ship. We now know that traversing a large body of water like the Atlantic, the winds will be very different at various spots on your journey. So you might reasonably extrapolate a couple hundred miles = ~90 minutes into your future, but not farther.
If your preflight planning weather predictions turned out to be pretty close to what really happened out there, you’ll probably find your target. If not, either you land in Ireland instead of Paris, or you get to swim for a few minutes before drowning.