This would be almost impossible for a few reasons. But hey, impossible makes for an interesting story, so let's work through it with a few creative stretches here and there.
The Chinese were interested in tribute. It's essentially a racket. You give us horses, gold, and other wealth, as well as whatever other curiosities you have laying around, and we don't kill you. Their focus politically was also on the maintenance of their own empire and the management of the millions of people under their sway. One of the reasons China never had a large maritime empire like England did is because there was no need: they had a huge land area and population in China to control and rule, and they had access to lucrative trade routes, as well as tribute from nearby countries to bolster their wealth. England is a fairly resource poor island with a low population, so they had to go around the world conquering other nations and stealing their resources. China didn't.
So for this young Aztec prince to get what he wants he's going to have to find a way to:
- Cross the Pacific ocean.
- Enter into tributary status with the Ming dynasty
- Absorb enough of their technology and culture.
- Bring it back to Mexico and spread this new found thought to your people.
With the technology of the Aztec at the time, crossing the Pacific ocean would be an incredible feat. I know Polynesians were able, over many centuries, to colonize the Pacific Ocean, but that took a long time, probably a ton of people died, and people were just landing and settling where they could with no real plan. However, this wouldn't make for an interesting story, so give the Prince a vision of a massive boat. After all, it's prophecy, right?
So, now we have our Aztecs and a boat large enough to house 100 of them decently with food stores. To meet the Ming emperor you need gifts. Yes, it is hard to buy gifts for the emperor who has everything. Luckily, we're talking about the YongLe emperor, so he should be sated with some Aztec gold craftsmanship as well as other unique gifts the Aztecs could fashion. It'd also be good to bring seeds of certain crops to offer the Chinese fruits and vegetables they would not have had at the time. The Chinese have pretty much all they need, so anything unique the Aztecs could muster would be much appreciated. Also, and this is big, bring documentation from the king, maybe a seal, to show that the envoy is not a bunch of nobodies, but rather, important people who can speak for the Aztec king. I forgot his name, but there was some Japanese dude who wanted to enter into a tributary relationship with China for silver, and he wasn't high enough of a title, so the Chinese rebuffed him until he was the proper title. It took 40 years.
Now, let's say by some miracle the Aztec sailors make it across the Pacific and they land somewhere in Indonesia or the Phillipines. They trade with locals to get food and fresh water, and by figuring out the local languages, they are guided to Ming China by way of Vietnam. Literally no one in this part of the world would know who the Aztecs were, and why they should give them the time of day, so the sailors are going to have to play up the Aztec empire as if it is somehow analogous to the Chinese empire, just in a different direction. Maps would help, artwork showing great Aztec armies marching would help, and a couple good storytellers with a linguist to help communicate. It's going to be a hard sell to convince the locals you're not a bunch of crazy people off of a boat, and that there really is a mystical kingdom far to the east where there are half jaguar-half human warriors who feast on the remains of their fallen enemies in battle (or whatever story you want to make up that would fit).
By pure luck, you run into Zheng He who is going off on one of his expeditions to show the world just how much bling China has (spoiler alert: it's a lot). He gives your envoy a tally that gives them the right to journey to the capital. Now, the Aztec envoy is going to have to survive the journey all the way to Beijing (YongLe moved shop from Nanjing), kowtow to the emperor, show off your great Aztec treasures, and enter into a tributary relationship with him.
The next problem is that the Yong Le emperor is mostly busy dealing with a China that's just come from a rebellion. The dude is going to be slightly paranoid because there should be hundreds of people plotting and scheming to install themselves into power. He might not have time to consider entering into a tributary relationship with a civilization that's over 13,000 km away. On the other hand, it's no skin off his nose if he accepts the gifts, lets his scholars learn about Aztec culture, and let's you stay in Beijing long enough to grow corn, yams, potatoes, tomatoes, and avocados. The Chinese may not like the avocado so much, but the rest of the vegetables will be put to good use.
What would be beneficial from the Aztec perspective is for the linguistically inclined person in the group to learn Chinese and get his hands on some ship building manuals, self defense manuals for martial arts, and any texts that deal with mathematics. The Chinese would probably push the Confucian stuff as well, but that is largely useless philosophy. The Ming dynasty would focus on it too much and then stagnate. The Aztecs are interested in the gun powder and the city planning, not on the correct rites.
Now, the hard part, getting back. If you got on the emperor's good side then you can ask for him to send Zheng He to Mexico. Zheng He had convinced the emperor to send out ships to far away lands, so if Mexico doesn't count I don't know what does. Zheng He is new found friend, and after a few years, the Chinese build some awesome ships that can take on the Pacific. Whoever is alive of the Aztec envoy boards the boat filled with Chinese knowledge of astronomy, mathematics, ship building, city planning, warfare, agriculture, and science, ready to take it back to the Aztecs.