Appropriate word for a young person who behaves like a cynical old person?

A young person who demonstrates wisdom and maturity beyond their years is often called an old soul. It comes from the belief that some reincarnated souls retain a measure of the wisdom and character developed in previous incarnations. These days, even those who don't share in the actual belief find use for the phrase in normal conversation. I'd be perfectly comfortable referring to the boy you describe with this term.

Sclerotic: an inability to adapt. For example,

That boy Tim is a sclerotic kid — age 13 going on 65.

He sounds like a fuddy-duddy

one that is old-fashioned, unimaginative, or conservative [Merriam-Webster]

The term is not limited to children, but is often applied to someone who seems old beyond their years.

You also might consider fogey

an extremely fussy, old-fashioned, or conservative person (esp in the phrase old fogey) [Collins]

In this case, he's a young fogey.

You also might consider relic

(informal) an old or old-fashioned person or thing [Collins]

And finally, throwback

one that is suggestive of or suited to an earlier time or style: his manners were a throwback to a more polite era [Merriam-Webster]

The lad you speak of has an anachronistic perspective.

Anachronistic may seem to be a stretch here, but when you think about it, an anachronistic perspective (attitude, outlook, way of looking at things) is a perspective that is somehow out of order chronologically.

Very often we think of anachronistic thinking as backward-looking, from the perspective of the present, such as the existence today of "colored only" drinking fountains or "colored only" restrooms, which are things of the past. Why couldn't a forward-looking perspective be anachronistic in the sense that it is ahead of its time?

Someone feel free to correct me if I'm off base.