I'm going to assume you're not talking about something that is genetically engineered for this purpose (otherwise I think the answer is pretty straightforward).
The closest analogue to your dragon would be a jet-powered, pressurized Cessna 172.
You're going to have a few hurdles to overcome here.
A Cessna 172 has a normal service ceiling of approximately 4000m (13000ft). Much above that, and you'd need to pressurize the plane for the passengers. Even if we assumed we could create a creature that could operate at over twice this height, its passengers certainly would not.
A system like this is really hard to explain evolutionarily. There are three pieces here: (1) the combustion chamber, (2) the fuel, and (3) the ignition source
A rudimentary combustion chamber could be a squirting sack of some sort. Our best example in real life is the octopus (and related animals). They use a powerful squirt of water to quickly move through the ocean. This gives us a possible explanation for how these muscles developed.
Once we get the sack, we need something inside it. Obviously, for a flying creature, filling a large air bladder, and then expelling it through a muscle controlled nozzle would give our create an evolutionary advantage in catching prey or escaping predators. And since we already have an evolutionary analog for this, I'm not going into the details here.
Like you said, a possible fuel source is methane. Conceivably, if our creature is an omnivore, it may produce methane through its diet. An evolutionary advantage may lead to this being stored in the air bladder instead of being directly expelled. Eventually, all the creature's produced methane would be in the air bladder.
At some point, once the creature can produce its own "air", it may evolve a storage capacity (basically, a second air bladder), which would allow it to refill the primary air bladder more quickly.
What I believe I have shown is an evolutionary pathway for a bird-like creature to develop multiple air bladders, of which at least one is used for supplemental propulsion, where the air is in part methane produced by the creature.
However, that doesn't quite answer the fuel part.
Methane is approximately 55MJ/kg, or 36.4kJ/L. Contrast that to jet fuel, which, while only 46MJ/kg, is 37.4MJ/L (note the megajoule unit in the second one). It's got approximately 1000x more volumetric energy density.
A cow produces somewhere between 100 to 500L of methane a day. Assuming we could reliably produce 500L of methane, this is really only half a liter of jet fuel worth of energy. Assume that is proportional to the mass of the animal (an adult cow weights approx. 750kg), that's about 0.7 mL/kg. Even if you increased that to 10 mL/kg (assuming our creature is super efficient about producing methane) and you'd still need a creature that weighs 21 tonnes to fill a Cessna 172 with fuel every day. For comparison, elephants weigh 6 tonnes, and the largest flying creature to ever live only weighed 250kg - one third of a cow.
So, you'd need some other way of fueling this beast. If you want to use gases, you'd need to make your creature massive to handle the volume (or massive with muscles to create the needed pressure), or you'd have to use something else.
Enough jet fuel would require 171kg (half the weight of or weightiest flier). Even compressed hydrogen would require 55kg worth of fuel (to reach the same energy). But, of course, what good is fuel without...
Now we get to the crux of why this will never work. Since something as complicated as a highly balanced rotating shaft will never exist in the organic world, we need an alternative. The only two non-rotating jet engines are the pulsejet and the ramjet.
Ramjets are very efficient--but only at about mach 3 or above. At the very least, you need supersonic airflow to make them work. So, unless you have a way for your creature to flap its way past mach 1, a ramjet would never work.
Pulsejets give very poor compression, and are not very fuel efficient. This would require even more fuel. They are also very loud. However, I could see an evolutionary explanation (though we are now straining reality) for this.
Now, even for a pulsejet to work, we would need our creature to be able to carefully add in air and fuel, spark the mixture, and then withstand the intense heat of combustion.
The first step is not impossible, the second step is highly improbably, and the third step is almost impossible. How is a dragon supposed to be able to create a combustion chamber inside of their air bladder?
All in all, I think this is an impossible task. You are trying to make a natural creature that evolved both flight with wings, and with flaming jet powered farts. Not going to happen.
TL;DR - You can't power a dragon with fartjets.