If you were to fly a plane while traveling the same direction as the sun, could you technically maintain your position at solar noon?

You’d have to travel pretty fast if you were over the equator, but if you were at higher or lower lattitudes not so much.
Yes, you can go that fast, but you’ll need to stop and get fuel once in awhile. That will slow you down.
And everytime you cross the International Date Line in the Pacific your day will jump forward or back depending on whether you fly East or West.
I once flew from Sydney, Australia, on a non-stop flight to San Diego, California.
I left on Tuesday at 10 AM. On the plane I saw night come and go and the sun rise.
I landed in California at 9:30 AM on Tuesday (the same day I left) and it was now 30 minutes before the time I left Australia. Go figure. It’s that darn International Date Line in the Pacific that does that to you.