They’re both really really close to perfect, and any native speaker would understand exactly what you’re saying with no problem, but I’d say them like this:

“I won’t be at my computer much today, I have to be somewhere.”

and

“It’s incredible how time flies! It will be Christmas soon!”

Close...

"I won't be able to use much the computer today. I have to go to a place."

Should be:

"I won't be able to use the computer much today, I have to go to a place."

Alternately:

"I won't be able to use the computer very much today, I have to go to a place."

Much (or very much) is modifying the action of using the computer, so it comes after.

"It's incredible how flies the time! It will be Christmas soon!"

Same issue here...

"It's incredible how time flies! It will be Christmas soon!"

The idiom is "time flies", flies, the modifier, comes after. "The" isn't needed and can be left out.

English has a LOT of weird rules, but there are good books that explain EVERYTHING:

Strunk and White's "Elements of Style" is kind of the gold standard and it's old enough now that it's public domain:

https://www.gutenberg.org/ebooks/37134

While not free, I'd also suggest the Borzoi Handbook for Writers:

https://www.amazon.com/dp/0394332016

"I won't be able to use many computers today." See how much/many in this position would indicate a multiple of computer? As though you could only use a few. But you're not wanting to count the number of computers but the amount of time. In the original, "the" before "computer" indicates it's singular so that the original sentence is simply not grammatical. What you probably mean is: "I won't be able to use the computer much today."

(Edit: I think in older English you could also put "much" earlier in the sentence: "I won't much be able to use the computer today." Makes me think of "Tell me where is gandalf, for I much desire to speak with him." https://youtu.be/uE-1RPDqJAY?t=74)

For the second sentence, swap the verb and the noun. I don't know why. "It's incredible how the time flies!" Putting the verb before the time might fit in a question: "How flies the time? Like an arrow!"

No, those are not correct.

If I saw either of those, I would assume that you started learning English recently. (Which is fine, it just means more work for us to understand each other)

As other comments have said the meaning is mostly clear but it's not how native speakers talk.

Something I've noticed is that when South Asian people refer to someone by their first and last name, they always say Smith, John, instead of John Smith.