This is absolutely doable, but with some slight bending of the rules. All that is really required, as far as your prompt goes, is that the character discovers and flys a personal flying machine away from the temple for an indeterminate distance.
Let's start with the most basic approach by giving our hero a hang glider or paraglider. Both of these use few moving parts (mostly pulleys and tensioning elements), and use no fluids or fuel.
The only limit to flight-worthness is the breakdown rate of the various airframe elements: Nylon, Kevlar (and other amarids), foam in the harness/seat, and potentially the epoxy/plastic composing the CFRP/GFRP spars of a hang glider are at risk. I suspect that if these are stored away from the atmosphere (say in an Argon-filled container), they will be perfectly preserved.
This gets our hero flying with a glide ratio of ~ 10:1 (if not just falling with style).
Storage Requirements: 2m^3 Argon-filled vault.
Maintenance Required: None
Ok, let's add the requirement that our hero must fly in sustained level flight with the vehicle under its own power. We've already solved the issues with the airframe surviving the time frame required, let's add power:
Hang gliders and paragliders become self-powered by basically adding a 2-stroke engine and maybe a set of tricycle wheels. The new list of dependencies is everything that is needed by the engine: Oil, fuel, lubricants, spare parts such as bearings and O-rings.
If present, an electronic ignition system may present a challenge, as any driving electronics would have to address the following:
- Tin Whiskering, where soldered parts develop structures that short to other legs on ICs.
- Silicon Dopant migration/diffusion.
- EEPROM or Flash data retention.
- Capacitors derating due to breakdown of dielectric material.
Glow plug or magneto ignition methods probably are the safest bet in light of the above list.
If our hero can follow instructions to produce reasonable substitutes for the various perishable fluids listed above, the motor may well be able to operate. This results in our hero flying perhaps 300 kilometers at 15 m/s (180 miles at 30 knots).
Storage Requirements: 4m^3 Argon-filled vault.
Maintenance Required: All fluids for the motor.
So you want our hero to fly in a more substantial aircraft for a much further distance? Let's try a self-launching* jet-powered sailplane! I present to you the JS1:
This thing is made out of carbon fiber (CFRP), along with various other elements like rubber handles, polycarbonate canopy, adhesives holding possibly-perishable upholstery in place, and so on. It all can be disassembled and placed in a transport trailer like this:
Typical Sailplane Trailer
So, if the aircraft is stored in the trailer, and said trailer is placed in a inert-gas vault (again, Argon), there is a good bet it will be functional in a few hundred years. Note that the jet nacelle extension/retraction/ignition system is battery powered, so the battery will have to be duplicated or compensated for when discovered by our hero.
As with solution 1.5 above, the fuel used in the jet needs to be produced by our hero from local sources. Jet engines can be fairly hardy, able to burn various types of fuel, I'll leave it to others more familiar with the chemistry to explain what to produce and how. On that note: Can kerosene last many hundreds of years?
The result of this solution: Easily 1000 km in a single flight at 40 m/s (600 miles at 80 knots).
Storage Requirements: 40m^3 Argon-filled vault.
Maintenance Required: Substantial.
A possibility simpler way to preform the mission is to use an electric aircraft. As an electrical engineer, I would love for this to be the prime solution, but it is so much more difficult than the combustion methods above, and because of this fact: Modern lithium batteries, and most any batteries really, tend to self-discharge and self-degrade. To last hundreds of years is a hard problem, and to have our hero make his own is laughable.
Solar power is another option, which could be combined with the battery option above or as the sole power source. However, this method would be on a higher level of difficultly than the jet-powered sailplane explained above; much difficulty to be had with solar cells having the silicon junctions diffuse over hundred of years, ditto for MPPT circuitry, etc.
In summery: If it wasn't for the fact that batteries degrade, capacitor dielectrics fail and silicon dopants bleed, electric would be wonderful, as the only moving parts would be the electric motor shaft and bearings.
I've ignored EMP issues here as I'm assuming the temple is far from the devastation and all suspectable artifacts are stored in Faraday Cages. Further, most modern control electronics are fairly hardy, if properly designed and following correct wiring harness practices.
*The JS1 isn't actually intended to self-launch, but no reason why the story can't have a jet-powered aircraft that can.