Hero does not actually need his Tinker powers to figure out how to use that. What songs are there?

Perception (2x)

Perception (3.6x, read warnings)

Skip sleep (read warnings)

Ooomph

Sleep (standard)

Sleep (Millennia, read warnings)

Crop growth

Wind

Rain

Propulsion

Warmth

Focus

Navigation

Okay he wants to read the warnings first.

Perception 3.6x: This alters the mind but not the body; it will seem noticeably slower to respond to commands, and some people report feeling trapped in their body as it is markedly slow to react to their attempts to move it. When the song stops most people experience a hangover period of a few minutes to a few hours, varying by duration spent listening to the songs; during that time period you will be nauseous, clumsy, and have a migraine. It is common for this reason to drop into accelerated perception for weeks at a time so as to minimize the inconvenience of dropping back out; if you do this, please carefully monitor your diet to ensure you are eating enough or else heal yourself regularly. 

Skip sleep: Humans should not use this to go more than 96 hours without sleep; between 150 and 200 hours humans who have not slept will abruptly collapse and be impossible to wake for around ten hours (and, if not woken, will sleep for about twenty). Elves should not use this to go more than 240 hours without sleep; between 260 and 380 hours Elves who have not slept will abruptly collapse and be impossible to wake for about four hours (and, if not woken, will sleep for about twelve.) Species with sleep needs comparable to human tend to have comparable to human reactions.

Sleep (Millennia): This will put everyone who can hear the song to sleep until several hours after the song stops playing. They will not be possible to wake for at least half that time.

"What's 'Ooomph' and 'Propulsion' and 'Focus' and 'Navigation'?"

"Oomph makes - whatever you're doing - stronger, just sort of a force multiplier thing. It interacts weirdly with other magic systems, we can test, but it probably makes people stronger and might extend the range of their powers and if their power is kind of -blasty - then it'd probably be more blasty. Focus makes you not lose your train of thought but it also means you won't notice a siren and stuff. Propulsion makes boats go faster, it probably has a bit of an effect on cars too but not much of one, it's fun to stack with your own flight if you've got flight but not especially useful. Navigation points to magnetic north if you're on a planet."

"I kinda wanna test all of these and wow you're right I'm super awake but anyway if I go test these I'm not gonna be helping you with things."

"Yeah, we were going to figure out how to make fancy future vehicles and computers on your current tech base or one I could demon you."

"Right. Fancy future vehicles and computers. They're not much my speciality but can do. Um, what do we need? Should I show you stuff we have or...?"

"What is your specialty, we could start there -"

"Well, there's a bunch of future stuff. I have my jetpack and a disintegration gun and a stun gun and sonic weapons and I'm really good at batteries and other power sources!"

"Oooh, awesome, we can totally start with batteries." And he makes some future batteries in various states of dismantlement - "personal computers that fit in your pocket are a thing by ten years from now, ones like mine a century after that, and battery life's the main reason they're not even tinier -"

He looks at the batteries and declares, "I can do better than that."

"Awesome. So then the question is whether I can duplicate your stuff -"

He can!

He will cackle with glee and fly around the Dome ship and then collapse on the grass giggling.

Hero will grin and giggle, too.

In that case they can forget duplicating the future and just figure out how to do better than them off Tinker powers. WIth the supplies, can Hero put together a 2010 cell phone? A 2020 cell phone? A 2030 thumb-sized computing device? A chip-locked computer?

Yes, yes, maybe, he will try.

"Should I try these now? Could take hours."

"I don't know! I want your world to have all the awesome stuff but I have no idea how to make that happen fastest -"

"Well I don't either, that's what all the PRT whizzes are gonna help us figure out. I don't think we can do it overnight, though."

"Probably not. ...but I could totally show you all the cool stuff tonight, wanna at least see it?"

"Yes!!!!!"

"This is a Dome ship, aliens called Andalites made them, I made one because it has the best artificial gravity in the multiverse, here's a scale model, here's a blueprint -"

Hero drinks it up. "You know," he says, "way my power works is I get inspiration for stuff. I don't really understand what I'm doing. So I make it up by trying to figure out as much as I can on my own, you have no idea how good it feels to see an actual blueprint that makes sense and regular people could read."