You wrote:
In your hand every day is a device that a thousand times outperforms
the computer that drove Apollo 11 landing on the Moon, yet hardly one
acknowledges it, let alone fully appreciate it.
You can put the phrase in either of two places:
In your hand every day is a device that outperforms by a thousand
times the computer that drove Apollo 11's landing on the Moon...
In your hand every day is a device that outperforms the computer that drove Apollo 11's landing on the Moon by a thousand times...
The former is stylistically clearer in that it does not separate the adverbial modifier from its verb, outperforms, and does not admit the confusion where the adverbial phrase might be thought to modify the verb drove.
For the sake of thoroughness, you could also put it here, but I would not advise that you do so:
In your hand every day is a device that by a thousand
times outperforms the computer that drove Apollo 11's landing on the Moon...
A cataphoric adverbial modifier in the form of a prepositional phrase is somewhat more difficult to parse.
However, sometimes you have no other clearly better place to put it:
... and with her eyes shut struck the suspended papier-mache donkey, causing the goodies inside to spill out onto the ground.
... and struck the suspended papier-mache donkey with her eyes shut, causing the goodies inside to spill out onto the ground.
.. and struck with her eyes shut the suspended papier-mache donkey, causing the goodies inside to spill out onto the ground.