I have a proof of a result in number theory, shorter than the existing proofs. Can such results be published and how?

As @Nat says, do post in on arXiv. Many journals have a "Notes" section for short results like yours, which people would find interesting, especially if they were teaching similar results. The MAA Monthly does this, e.g. After you have the proof up on arXiv, submit it to the "Notes" section of an appropriate journal, following the specific directions for that section. At least one professional mathematician will look at it, so you'll get some vetting. Have a look at the Missouri Journal of Mathematical Sciences.

Can such results be published and how?

“Such results” certainly can be published, if by “such results” you mean genuinely new proofs of well-known theorems, especially if they contain a new and interesting idea (rather than being a trivial modification or variant of an existing proof), and especially if they are shorter than existing proofs, although that is not a necessary condition for a new proof to be publishable.

Such new proofs are published quite frequently. For example, when I was a graduate student I published a new proof of Stirling’s formula. It was published in the American Mathematical Monthly, a respectable journal that often publishes papers in this category.

Should I publish my result?

As others have said, you can at the very least write up your proof as a paper and submit it to arXiv to make it available to the community. And you can try to publish it in a journal - if it’s interesting, well-written and novel, I think you have a good chance of getting it published somewhere.

If so, as I'm not affiliated with any Research Institute how do I do that?

See this question for some suggestions.